Moment of Inertia

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I
Class 11 Physics - Rotation

Moment of Inertia

Master moment of inertia, radius of gyration, parallel axis theorem, perpendicular axis theorem, MOI of rod, ring, disc, cylinder, sphere, numericals and PYQs.

CBSENEETJEE MainJEE AdvancedIBIGCSEA-Level
01

Moment of Inertia

Moment of inertia is rotational inertia. It measures how difficult it is to change rotational motion about a chosen axis.

I = Σmr2I = ∫r2 dm
  • Depends on mass distribution and axis.
  • SI unit: kg m2.
  • Dimensions: ML2.
  • Mass farther from axis gives larger I.
disc axes
02

Radius of Gyration

Radius of gyration is the equivalent distance at which the whole mass can be imagined concentrated to give the same moment of inertia.

I = Mk2k = √(I/M)

k is not necessarily the actual radius; it is an inertia-equivalent distance.

I = MR²
03

Parallel Axis Theorem

Moment of inertia about any axis equals MOI about a parallel COM axis plus Md2.

I = Icm + Md2

d is distance between parallel axes.

Solved example: rod centre to end
Iend=ML2/12+M(L/2)2=ML2/3.
d
04

Perpendicular Axis Theorem

For a plane lamina, moment of inertia about perpendicular axis equals sum of moments about two mutually perpendicular in-plane axes.

Iz = Ix + Iy

Applicable only for plane lamina. Do not apply directly to 3D bodies.

Solved example: disc diameter
For disc, Iz=MR2/2 and Ix=Iy. Hence diameter I=MR2/4.
Iz = Ix + Iy
05

MOI of Rod

A uniform rod has different moment of inertia depending on whether the axis passes through centre, end, or another point.

Icentre = ML2/12Iend = ML2/3I = Icm + Md2

Integration: I=∫x2(M/L)dx.

centre and end axes
06

MOI of Ring

For a thin ring, all mass lies at distance R from the central axis.

Icentral = MR2Idiameter = MR2/2

Diameter result follows from perpendicular axis theorem.

I = MR²
07

MOI of Disc

For a uniform disc, mass is spread from centre to radius R, so central MOI is smaller than a ring.

Icentral = MR2/2Idiameter = MR2/4Itangent = 3MR2/2
disc axes
08

MOI of Cylinder

Solid, hollow and thick hollow cylinders differ because mass distribution is different.

Solid Cylinder

I = MR2/2

Hollow Cylinder

I = MR2

Thick Hollow Cylinder

I = 1/2 M(R12 + R22)
09

MOI of Sphere

A hollow sphere has larger MOI than a solid sphere for same M and R because more mass is farther from axis.

Solid Sphere

I = 2MR2/5

Hollow Sphere / Shell

I = 2MR2/3
solid vs hollow
10

Hollow vs Solid Body Comparison

For same mass and radius, hollow bodies generally have larger MOI because mass is farther from axis.

BodySolid MOIHollow MOIAxisExam Tip
CylinderMR²/2MR²Symmetry axisHollow larger
Sphere2MR²/52MR²/3DiameterShell larger
Disc/RingDisc MR²/2Ring MR²Central axisRing is like hollow disc
Thin shell ideasMass spread insideMass near surfaceGiven axisFarther mass means larger I
solid vs hollow
I = MR²
11

Important Formula Table

Complete formula sheet for quick revision.

ObjectAxisMoment of InertiaRadius of GyrationImportant Note
RodCentreML²/12L/√12Perpendicular to rod
RodEndML²/3L/√3Parallel axis
RingCentralMR²RAll mass at R
RingDiameterMR²/2R/√2Plane lamina
DiscCentralMR²/2R/√2Solid lamina
DiscDiameterMR²/4R/2Perpendicular axis
DiscTangent3MR²/2R√(3/2)Parallel axis
Solid cylinderSymmetryMR²/2R/√2Same as disc stack
Hollow cylinderSymmetryMR²RShell
Thick hollow cylinderSymmetry1/2M(R1²+R2²)√[(R1²+R2²)/2]Use both radii
Solid sphereDiameter2MR²/5R√(2/5)3D body
Hollow sphereDiameter2MR²/3R√(2/3)Shell larger

Searching for a Physics Tutor? If Moment of Inertia, Radius of Gyration, Axis Theorems or NEET/JEE numericals are not clear, contact Kumar Sir.

Phone: +91-9958461445 | Email: kumarsirphysics@gmail.com | Website: kumarphysicsclasses.com

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12

High-Quality Numericals

Solved bank covering basic MOI, axis theorems, standard bodies, hollow-solid comparison and composite bodies.

1. CBSE point mass: Find I of 2 kg mass at 3 m from axis.
disc axesQuestion: Find I of 2 kg mass at 3 m from axis.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=mr²=18 kg m².
Final Answer: I=mr²=18 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
2. NEET rod centre: Rod M=3 kg, L=2 m about centre.
centre and end axesQuestion: Rod M=3 kg, L=2 m about centre.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=ML²/12=1 kg m².
Final Answer: I=ML²/12=1 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
3. JEE Main rod end: Rod M=3 kg, L=2 m about end.
centre and end axesQuestion: Rod M=3 kg, L=2 m about end.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=ML²/3=4 kg m².
Final Answer: I=ML²/3=4 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
4. JEE Advanced parallel axis: Disc M,R about tangent axis.
dQuestion: Disc M,R about tangent axis.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=MR²/2+MR²=3MR²/2.
Final Answer: I=MR²/2+MR²=3MR²/2.
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
5. IB ring: Ring M=2 kg,R=0.5 m central I.
I = MR²Question: Ring M=2 kg,R=0.5 m central I.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=MR²=0.5 kg m².
Final Answer: I=MR²=0.5 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
6. IGCSE cylinder: Solid cylinder M=4,R=0.2. Find I.
Question: Solid cylinder M=4,R=0.2. Find I.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=MR²/2=0.08 kg m².
Final Answer: I=MR²/2=0.08 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
7. A-Level sphere: Solid sphere M=5,R=0.3. Find I.
solid vs hollowQuestion: Solid sphere M=5,R=0.3. Find I.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=2MR²/5=0.18 kg m².
Final Answer: I=2MR²/5=0.18 kg m².
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
8. Radius of gyration: If I=8 and M=2, find k.
disc axesQuestion: If I=8 and M=2, find k.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: k=sqrt(4)=2 m.
Final Answer: k=sqrt(4)=2 m.
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
9. Perpendicular theorem: Disc Iz=10. Find diameter I.
Iz = Ix + IyQuestion: Disc Iz=10. Find diameter I.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: Ix=Iy=5.
Final Answer: Ix=Iy=5.
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
10. Composite: Ring and disc same M,R coaxial. Find total I.
I = MR²Question: Ring and disc same M,R coaxial. Find total I.
Diagram: shown above.
Given: Identify mass, radius/length, body type and axis.
Formula: I=Σmr², I=∫r²dm, I=Icm+Md², Iz=Ix+Iy, and I=Mk².
Calculation: I=MR²+MR²/2=3MR²/2.
Final Answer: I=MR²+MR²/2=3MR²/2.
Exam Tip: Draw the axis before selecting formula.
Common Mistake: Using central-axis formula for tangent or diameter axis.
13

NEET Question Bank

50 NEET-style MCQs. Authentic years are not invented.

1. NEET Exam-style Question: Moment of inertia of point mass m at distance r is: A mr B mr² C m/r D r/m
disc axesCorrect Answer: B. I=mr2.
Detailed Explanation: This tests definition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
2. NEET Exam-style Question: SI unit of moment of inertia is: A kg m B kg m² C N m D J
disc axesCorrect Answer: B. kg m2.
Detailed Explanation: This tests unit. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
3. NEET Exam-style Question: If I=Mk², k equals: A I/M B sqrt(I/M) C IM D M/I
disc axesCorrect Answer: B.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
4. NEET Exam-style Question: Uniform rod about centre has I: A ML²/12 B ML²/3 C MR² D MR²/2
centre and end axesCorrect Answer: A.
Detailed Explanation: This tests rod centre. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
5. NEET Exam-style Question: Uniform rod about end has I: A ML²/12 B ML²/3 C ML²/2 D ML²
centre and end axesCorrect Answer: B.
Detailed Explanation: This tests rod end. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
6. NEET Exam-style Question: Thin ring about central axis has I: A MR² B MR²/2 C 2MR²/5 D MR²/4
I = MR²Correct Answer: A.
Detailed Explanation: This tests ring. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
7. NEET Exam-style Question: Uniform disc about central axis has I: A MR² B MR²/2 C MR²/4 D 2MR²/5
disc axesCorrect Answer: B.
Detailed Explanation: This tests disc. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
8. NEET Exam-style Question: Solid cylinder about symmetry axis has I: A MR² B MR²/2 C 2MR²/3 D ML²/12
Correct Answer: B.
Detailed Explanation: This tests solid cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
9. NEET Exam-style Question: Solid sphere about diameter has I: A 2MR²/5 B 2MR²/3 C MR² D MR²/2
solid vs hollowCorrect Answer: A.
Detailed Explanation: This tests sphere. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
10. NEET Exam-style Question: For same M,R, hollow sphere has larger I because mass is: A nearer axis B farther from axis C zero D same point
solid vs hollowCorrect Answer: B.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
11. NEET Exam-style Question: Moment of inertia of point mass m at distance r is: A mr B mr² C m/r D r/m
disc axesCorrect Answer: B. I=mr2.
Detailed Explanation: This tests definition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
12. NEET Exam-style Question: SI unit of moment of inertia is: A kg m B kg m² C N m D J
disc axesCorrect Answer: B. kg m2.
Detailed Explanation: This tests unit. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
13. NEET Exam-style Question: If I=Mk², k equals: A I/M B sqrt(I/M) C IM D M/I
disc axesCorrect Answer: B.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
14. NEET Exam-style Question: Uniform rod about centre has I: A ML²/12 B ML²/3 C MR² D MR²/2
centre and end axesCorrect Answer: A.
Detailed Explanation: This tests rod centre. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
15. NEET Exam-style Question: Uniform rod about end has I: A ML²/12 B ML²/3 C ML²/2 D ML²
centre and end axesCorrect Answer: B.
Detailed Explanation: This tests rod end. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
16. NEET Exam-style Question: Thin ring about central axis has I: A MR² B MR²/2 C 2MR²/5 D MR²/4
I = MR²Correct Answer: A.
Detailed Explanation: This tests ring. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
17. NEET Exam-style Question: Uniform disc about central axis has I: A MR² B MR²/2 C MR²/4 D 2MR²/5
disc axesCorrect Answer: B.
Detailed Explanation: This tests disc. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
18. NEET Exam-style Question: Solid cylinder about symmetry axis has I: A MR² B MR²/2 C 2MR²/3 D ML²/12
Correct Answer: B.
Detailed Explanation: This tests solid cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
19. NEET Exam-style Question: Solid sphere about diameter has I: A 2MR²/5 B 2MR²/3 C MR² D MR²/2
solid vs hollowCorrect Answer: A.
Detailed Explanation: This tests sphere. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
20. NEET Exam-style Question: For same M,R, hollow sphere has larger I because mass is: A nearer axis B farther from axis C zero D same point
solid vs hollowCorrect Answer: B.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
21. NEET Exam-style Question: Moment of inertia of point mass m at distance r is: A mr B mr² C m/r D r/m
disc axesCorrect Answer: B. I=mr2.
Detailed Explanation: This tests definition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
22. NEET Exam-style Question: SI unit of moment of inertia is: A kg m B kg m² C N m D J
disc axesCorrect Answer: B. kg m2.
Detailed Explanation: This tests unit. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
23. NEET Exam-style Question: If I=Mk², k equals: A I/M B sqrt(I/M) C IM D M/I
disc axesCorrect Answer: B.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
24. NEET Exam-style Question: Uniform rod about centre has I: A ML²/12 B ML²/3 C MR² D MR²/2
centre and end axesCorrect Answer: A.
Detailed Explanation: This tests rod centre. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
25. NEET Exam-style Question: Uniform rod about end has I: A ML²/12 B ML²/3 C ML²/2 D ML²
centre and end axesCorrect Answer: B.
Detailed Explanation: This tests rod end. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
26. NEET Exam-style Question: Thin ring about central axis has I: A MR² B MR²/2 C 2MR²/5 D MR²/4
I = MR²Correct Answer: A.
Detailed Explanation: This tests ring. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
27. NEET Exam-style Question: Uniform disc about central axis has I: A MR² B MR²/2 C MR²/4 D 2MR²/5
disc axesCorrect Answer: B.
Detailed Explanation: This tests disc. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
28. NEET Exam-style Question: Solid cylinder about symmetry axis has I: A MR² B MR²/2 C 2MR²/3 D ML²/12
Correct Answer: B.
Detailed Explanation: This tests solid cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
29. NEET Exam-style Question: Solid sphere about diameter has I: A 2MR²/5 B 2MR²/3 C MR² D MR²/2
solid vs hollowCorrect Answer: A.
Detailed Explanation: This tests sphere. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
30. NEET Exam-style Question: For same M,R, hollow sphere has larger I because mass is: A nearer axis B farther from axis C zero D same point
solid vs hollowCorrect Answer: B.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
31. NEET Exam-style Question: Moment of inertia of point mass m at distance r is: A mr B mr² C m/r D r/m
disc axesCorrect Answer: B. I=mr2.
Detailed Explanation: This tests definition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
32. NEET Exam-style Question: SI unit of moment of inertia is: A kg m B kg m² C N m D J
disc axesCorrect Answer: B. kg m2.
Detailed Explanation: This tests unit. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
33. NEET Exam-style Question: If I=Mk², k equals: A I/M B sqrt(I/M) C IM D M/I
disc axesCorrect Answer: B.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
34. NEET Exam-style Question: Uniform rod about centre has I: A ML²/12 B ML²/3 C MR² D MR²/2
centre and end axesCorrect Answer: A.
Detailed Explanation: This tests rod centre. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
35. NEET Exam-style Question: Uniform rod about end has I: A ML²/12 B ML²/3 C ML²/2 D ML²
centre and end axesCorrect Answer: B.
Detailed Explanation: This tests rod end. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
36. NEET Exam-style Question: Thin ring about central axis has I: A MR² B MR²/2 C 2MR²/5 D MR²/4
I = MR²Correct Answer: A.
Detailed Explanation: This tests ring. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
37. NEET Exam-style Question: Uniform disc about central axis has I: A MR² B MR²/2 C MR²/4 D 2MR²/5
disc axesCorrect Answer: B.
Detailed Explanation: This tests disc. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
38. NEET Exam-style Question: Solid cylinder about symmetry axis has I: A MR² B MR²/2 C 2MR²/3 D ML²/12
Correct Answer: B.
Detailed Explanation: This tests solid cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
39. NEET Exam-style Question: Solid sphere about diameter has I: A 2MR²/5 B 2MR²/3 C MR² D MR²/2
solid vs hollowCorrect Answer: A.
Detailed Explanation: This tests sphere. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
40. NEET Exam-style Question: For same M,R, hollow sphere has larger I because mass is: A nearer axis B farther from axis C zero D same point
solid vs hollowCorrect Answer: B.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
41. NEET Exam-style Question: Moment of inertia of point mass m at distance r is: A mr B mr² C m/r D r/m
disc axesCorrect Answer: B. I=mr2.
Detailed Explanation: This tests definition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
42. NEET Exam-style Question: SI unit of moment of inertia is: A kg m B kg m² C N m D J
disc axesCorrect Answer: B. kg m2.
Detailed Explanation: This tests unit. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
43. NEET Exam-style Question: If I=Mk², k equals: A I/M B sqrt(I/M) C IM D M/I
disc axesCorrect Answer: B.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
44. NEET Exam-style Question: Uniform rod about centre has I: A ML²/12 B ML²/3 C MR² D MR²/2
centre and end axesCorrect Answer: A.
Detailed Explanation: This tests rod centre. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
45. NEET Exam-style Question: Uniform rod about end has I: A ML²/12 B ML²/3 C ML²/2 D ML²
centre and end axesCorrect Answer: B.
Detailed Explanation: This tests rod end. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
46. NEET Exam-style Question: Thin ring about central axis has I: A MR² B MR²/2 C 2MR²/5 D MR²/4
I = MR²Correct Answer: A.
Detailed Explanation: This tests ring. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
47. NEET Exam-style Question: Uniform disc about central axis has I: A MR² B MR²/2 C MR²/4 D 2MR²/5
disc axesCorrect Answer: B.
Detailed Explanation: This tests disc. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
48. NEET Exam-style Question: Solid cylinder about symmetry axis has I: A MR² B MR²/2 C 2MR²/3 D ML²/12
Correct Answer: B.
Detailed Explanation: This tests solid cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
49. NEET Exam-style Question: Solid sphere about diameter has I: A 2MR²/5 B 2MR²/3 C MR² D MR²/2
solid vs hollowCorrect Answer: A.
Detailed Explanation: This tests sphere. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
50. NEET Exam-style Question: For same M,R, hollow sphere has larger I because mass is: A nearer axis B farther from axis C zero D same point
solid vs hollowCorrect Answer: B.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
14

JEE Main Question Bank

50 difficult JEE Main-style questions on axis theorem, formulas, radius of gyration and composite MOI.

1. JEE Main Exam-style Question: Disc Icm=MR²/2. Find I about tangent in plane parallel to central axis.
dCorrect Answer: I=Icm+MR²=3MR²/2.
Detailed Explanation: This tests parallel axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
2. JEE Main Exam-style Question: Disc Iz=MR²/2. Find I about diameter.
Iz = Ix + IyCorrect Answer: Ix=Iy and Iz=Ix+Iy, so Ix=MR²/4.
Detailed Explanation: This tests perpendicular axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
3. JEE Main Exam-style Question: Ring central I=MR². Find k.
I = MR²Correct Answer: k=sqrt(I/M)=R.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
4. JEE Main Exam-style Question: Use parallel axis from rod centre to end.
centre and end axesCorrect Answer: Iend=ML²/12+M(L/2)²=ML²/3.
Detailed Explanation: This tests rod theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
5. JEE Main Exam-style Question: Thick hollow cylinder radii R1,R2 has I:
Correct Answer: I=1/2 M(R1²+R2²).
Detailed Explanation: This tests cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
6. JEE Main Exam-style Question: Two point masses m at distances r and 2r. Find I.
disc axesCorrect Answer: I=mr²+m(2r)²=5mr².
Detailed Explanation: This tests composite. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
7. JEE Main Exam-style Question: Disc central I=MR²/2. Tangent parallel axis?
dCorrect Answer: 3MR²/2.
Detailed Explanation: This tests disc tangent. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
8. JEE Main Exam-style Question: Ring central I=MR². Diameter I?
I = MR²Correct Answer: MR²/2.
Detailed Explanation: This tests ring diameter. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
9. JEE Main Exam-style Question: Ratio hollow sphere I to solid sphere I.
solid vs hollowCorrect Answer: (2/3)/(2/5)=5/3.
Detailed Explanation: This tests sphere comparison. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
10. JEE Main Exam-style Question: Solid cylinder k about axis.
Correct Answer: k=R/sqrt2.
Detailed Explanation: This tests solid cylinder k. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
11. JEE Main Exam-style Question: Disc Icm=MR²/2. Find I about tangent in plane parallel to central axis.
dCorrect Answer: I=Icm+MR²=3MR²/2.
Detailed Explanation: This tests parallel axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
12. JEE Main Exam-style Question: Disc Iz=MR²/2. Find I about diameter.
Iz = Ix + IyCorrect Answer: Ix=Iy and Iz=Ix+Iy, so Ix=MR²/4.
Detailed Explanation: This tests perpendicular axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
13. JEE Main Exam-style Question: Ring central I=MR². Find k.
I = MR²Correct Answer: k=sqrt(I/M)=R.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
14. JEE Main Exam-style Question: Use parallel axis from rod centre to end.
centre and end axesCorrect Answer: Iend=ML²/12+M(L/2)²=ML²/3.
Detailed Explanation: This tests rod theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
15. JEE Main Exam-style Question: Thick hollow cylinder radii R1,R2 has I:
Correct Answer: I=1/2 M(R1²+R2²).
Detailed Explanation: This tests cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
16. JEE Main Exam-style Question: Two point masses m at distances r and 2r. Find I.
disc axesCorrect Answer: I=mr²+m(2r)²=5mr².
Detailed Explanation: This tests composite. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
17. JEE Main Exam-style Question: Disc central I=MR²/2. Tangent parallel axis?
dCorrect Answer: 3MR²/2.
Detailed Explanation: This tests disc tangent. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
18. JEE Main Exam-style Question: Ring central I=MR². Diameter I?
I = MR²Correct Answer: MR²/2.
Detailed Explanation: This tests ring diameter. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
19. JEE Main Exam-style Question: Ratio hollow sphere I to solid sphere I.
solid vs hollowCorrect Answer: (2/3)/(2/5)=5/3.
Detailed Explanation: This tests sphere comparison. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
20. JEE Main Exam-style Question: Solid cylinder k about axis.
Correct Answer: k=R/sqrt2.
Detailed Explanation: This tests solid cylinder k. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
21. JEE Main Exam-style Question: Disc Icm=MR²/2. Find I about tangent in plane parallel to central axis.
dCorrect Answer: I=Icm+MR²=3MR²/2.
Detailed Explanation: This tests parallel axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
22. JEE Main Exam-style Question: Disc Iz=MR²/2. Find I about diameter.
Iz = Ix + IyCorrect Answer: Ix=Iy and Iz=Ix+Iy, so Ix=MR²/4.
Detailed Explanation: This tests perpendicular axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
23. JEE Main Exam-style Question: Ring central I=MR². Find k.
I = MR²Correct Answer: k=sqrt(I/M)=R.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
24. JEE Main Exam-style Question: Use parallel axis from rod centre to end.
centre and end axesCorrect Answer: Iend=ML²/12+M(L/2)²=ML²/3.
Detailed Explanation: This tests rod theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
25. JEE Main Exam-style Question: Thick hollow cylinder radii R1,R2 has I:
Correct Answer: I=1/2 M(R1²+R2²).
Detailed Explanation: This tests cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
26. JEE Main Exam-style Question: Two point masses m at distances r and 2r. Find I.
disc axesCorrect Answer: I=mr²+m(2r)²=5mr².
Detailed Explanation: This tests composite. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
27. JEE Main Exam-style Question: Disc central I=MR²/2. Tangent parallel axis?
dCorrect Answer: 3MR²/2.
Detailed Explanation: This tests disc tangent. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
28. JEE Main Exam-style Question: Ring central I=MR². Diameter I?
I = MR²Correct Answer: MR²/2.
Detailed Explanation: This tests ring diameter. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
29. JEE Main Exam-style Question: Ratio hollow sphere I to solid sphere I.
solid vs hollowCorrect Answer: (2/3)/(2/5)=5/3.
Detailed Explanation: This tests sphere comparison. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
30. JEE Main Exam-style Question: Solid cylinder k about axis.
Correct Answer: k=R/sqrt2.
Detailed Explanation: This tests solid cylinder k. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
31. JEE Main Exam-style Question: Disc Icm=MR²/2. Find I about tangent in plane parallel to central axis.
dCorrect Answer: I=Icm+MR²=3MR²/2.
Detailed Explanation: This tests parallel axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
32. JEE Main Exam-style Question: Disc Iz=MR²/2. Find I about diameter.
Iz = Ix + IyCorrect Answer: Ix=Iy and Iz=Ix+Iy, so Ix=MR²/4.
Detailed Explanation: This tests perpendicular axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
33. JEE Main Exam-style Question: Ring central I=MR². Find k.
I = MR²Correct Answer: k=sqrt(I/M)=R.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
34. JEE Main Exam-style Question: Use parallel axis from rod centre to end.
centre and end axesCorrect Answer: Iend=ML²/12+M(L/2)²=ML²/3.
Detailed Explanation: This tests rod theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
35. JEE Main Exam-style Question: Thick hollow cylinder radii R1,R2 has I:
Correct Answer: I=1/2 M(R1²+R2²).
Detailed Explanation: This tests cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
36. JEE Main Exam-style Question: Two point masses m at distances r and 2r. Find I.
disc axesCorrect Answer: I=mr²+m(2r)²=5mr².
Detailed Explanation: This tests composite. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
37. JEE Main Exam-style Question: Disc central I=MR²/2. Tangent parallel axis?
dCorrect Answer: 3MR²/2.
Detailed Explanation: This tests disc tangent. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
38. JEE Main Exam-style Question: Ring central I=MR². Diameter I?
I = MR²Correct Answer: MR²/2.
Detailed Explanation: This tests ring diameter. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
39. JEE Main Exam-style Question: Ratio hollow sphere I to solid sphere I.
solid vs hollowCorrect Answer: (2/3)/(2/5)=5/3.
Detailed Explanation: This tests sphere comparison. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
40. JEE Main Exam-style Question: Solid cylinder k about axis.
Correct Answer: k=R/sqrt2.
Detailed Explanation: This tests solid cylinder k. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
41. JEE Main Exam-style Question: Disc Icm=MR²/2. Find I about tangent in plane parallel to central axis.
dCorrect Answer: I=Icm+MR²=3MR²/2.
Detailed Explanation: This tests parallel axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
42. JEE Main Exam-style Question: Disc Iz=MR²/2. Find I about diameter.
Iz = Ix + IyCorrect Answer: Ix=Iy and Iz=Ix+Iy, so Ix=MR²/4.
Detailed Explanation: This tests perpendicular axis. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
43. JEE Main Exam-style Question: Ring central I=MR². Find k.
I = MR²Correct Answer: k=sqrt(I/M)=R.
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
44. JEE Main Exam-style Question: Use parallel axis from rod centre to end.
centre and end axesCorrect Answer: Iend=ML²/12+M(L/2)²=ML²/3.
Detailed Explanation: This tests rod theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
45. JEE Main Exam-style Question: Thick hollow cylinder radii R1,R2 has I:
Correct Answer: I=1/2 M(R1²+R2²).
Detailed Explanation: This tests cylinder. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
46. JEE Main Exam-style Question: Two point masses m at distances r and 2r. Find I.
disc axesCorrect Answer: I=mr²+m(2r)²=5mr².
Detailed Explanation: This tests composite. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
47. JEE Main Exam-style Question: Disc central I=MR²/2. Tangent parallel axis?
dCorrect Answer: 3MR²/2.
Detailed Explanation: This tests disc tangent. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
48. JEE Main Exam-style Question: Ring central I=MR². Diameter I?
I = MR²Correct Answer: MR²/2.
Detailed Explanation: This tests ring diameter. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
49. JEE Main Exam-style Question: Ratio hollow sphere I to solid sphere I.
solid vs hollowCorrect Answer: (2/3)/(2/5)=5/3.
Detailed Explanation: This tests sphere comparison. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
50. JEE Main Exam-style Question: Solid cylinder k about axis.
Correct Answer: k=R/sqrt2.
Detailed Explanation: This tests solid cylinder k. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
15

JEE Advanced Question Bank

50 advanced questions on integration, shifted axes, cut-outs, hollow-solid bodies and radius of gyration.

1. JEE Advanced Exam-style Question: Derive rod I about centre using dm=(M/L)dx.
centre and end axesCorrect Answer: I=∫x²dm from -L/2 to L/2 = ML²/12.
Detailed Explanation: This tests integration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
2. JEE Advanced Exam-style Question: A ring and disc same M,R are coaxially joined. Find total I.
I = MR²Correct Answer: I=MR²+MR²/2=3MR²/2.
Detailed Explanation: This tests composite bodies. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
3. JEE Advanced Exam-style Question: Rod axis at distance L/4 from centre. Find I.
dCorrect Answer: I=ML²/12+M(L/4)²=7ML²/48.
Detailed Explanation: This tests shifted axes. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
4. JEE Advanced Exam-style Question: Disc has small concentric hole removed. How compute remaining I?
disc axesCorrect Answer: Subtract removed disc MOI from full disc MOI using same axis.
Detailed Explanation: This tests cut-out MOI. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
5. JEE Advanced Exam-style Question: For same M,R, compare rolling rotational KE at same omega.
solid vs hollowCorrect Answer: Hollow has larger I, so larger rotational KE.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
6. JEE Advanced Exam-style Question: If I=3MR²/2, find k.
dCorrect Answer: k=R sqrt(3/2).
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
7. JEE Advanced Exam-style Question: Why perpendicular axis theorem cannot apply to solid sphere?
Iz = Ix + IyCorrect Answer: It is only for plane lamina.
Detailed Explanation: This tests perpendicular theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
8. JEE Advanced Exam-style Question: Thick cylinder with R1=R, R2=2R. Find I.
Correct Answer: I=1/2 M(R²+4R²)=5MR²/2.
Detailed Explanation: This tests thick shell. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
9. JEE Advanced Exam-style Question: What axes are allowed in parallel axis theorem?
dCorrect Answer: Parallel axes, one through COM, separated by d.
Detailed Explanation: This tests parallel theorem condition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
10. JEE Advanced Exam-style Question: Why ring has larger I than disc?
I = MR²Correct Answer: More mass is farther from axis.
Detailed Explanation: This tests mass distribution. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
11. JEE Advanced Exam-style Question: Derive rod I about centre using dm=(M/L)dx.
centre and end axesCorrect Answer: I=∫x²dm from -L/2 to L/2 = ML²/12.
Detailed Explanation: This tests integration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
12. JEE Advanced Exam-style Question: A ring and disc same M,R are coaxially joined. Find total I.
I = MR²Correct Answer: I=MR²+MR²/2=3MR²/2.
Detailed Explanation: This tests composite bodies. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
13. JEE Advanced Exam-style Question: Rod axis at distance L/4 from centre. Find I.
dCorrect Answer: I=ML²/12+M(L/4)²=7ML²/48.
Detailed Explanation: This tests shifted axes. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
14. JEE Advanced Exam-style Question: Disc has small concentric hole removed. How compute remaining I?
disc axesCorrect Answer: Subtract removed disc MOI from full disc MOI using same axis.
Detailed Explanation: This tests cut-out MOI. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
15. JEE Advanced Exam-style Question: For same M,R, compare rolling rotational KE at same omega.
solid vs hollowCorrect Answer: Hollow has larger I, so larger rotational KE.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
16. JEE Advanced Exam-style Question: If I=3MR²/2, find k.
dCorrect Answer: k=R sqrt(3/2).
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
17. JEE Advanced Exam-style Question: Why perpendicular axis theorem cannot apply to solid sphere?
Iz = Ix + IyCorrect Answer: It is only for plane lamina.
Detailed Explanation: This tests perpendicular theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
18. JEE Advanced Exam-style Question: Thick cylinder with R1=R, R2=2R. Find I.
Correct Answer: I=1/2 M(R²+4R²)=5MR²/2.
Detailed Explanation: This tests thick shell. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
19. JEE Advanced Exam-style Question: What axes are allowed in parallel axis theorem?
dCorrect Answer: Parallel axes, one through COM, separated by d.
Detailed Explanation: This tests parallel theorem condition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
20. JEE Advanced Exam-style Question: Why ring has larger I than disc?
I = MR²Correct Answer: More mass is farther from axis.
Detailed Explanation: This tests mass distribution. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
21. JEE Advanced Exam-style Question: Derive rod I about centre using dm=(M/L)dx.
centre and end axesCorrect Answer: I=∫x²dm from -L/2 to L/2 = ML²/12.
Detailed Explanation: This tests integration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
22. JEE Advanced Exam-style Question: A ring and disc same M,R are coaxially joined. Find total I.
I = MR²Correct Answer: I=MR²+MR²/2=3MR²/2.
Detailed Explanation: This tests composite bodies. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
23. JEE Advanced Exam-style Question: Rod axis at distance L/4 from centre. Find I.
dCorrect Answer: I=ML²/12+M(L/4)²=7ML²/48.
Detailed Explanation: This tests shifted axes. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
24. JEE Advanced Exam-style Question: Disc has small concentric hole removed. How compute remaining I?
disc axesCorrect Answer: Subtract removed disc MOI from full disc MOI using same axis.
Detailed Explanation: This tests cut-out MOI. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
25. JEE Advanced Exam-style Question: For same M,R, compare rolling rotational KE at same omega.
solid vs hollowCorrect Answer: Hollow has larger I, so larger rotational KE.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
26. JEE Advanced Exam-style Question: If I=3MR²/2, find k.
dCorrect Answer: k=R sqrt(3/2).
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
27. JEE Advanced Exam-style Question: Why perpendicular axis theorem cannot apply to solid sphere?
Iz = Ix + IyCorrect Answer: It is only for plane lamina.
Detailed Explanation: This tests perpendicular theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
28. JEE Advanced Exam-style Question: Thick cylinder with R1=R, R2=2R. Find I.
Correct Answer: I=1/2 M(R²+4R²)=5MR²/2.
Detailed Explanation: This tests thick shell. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
29. JEE Advanced Exam-style Question: What axes are allowed in parallel axis theorem?
dCorrect Answer: Parallel axes, one through COM, separated by d.
Detailed Explanation: This tests parallel theorem condition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
30. JEE Advanced Exam-style Question: Why ring has larger I than disc?
I = MR²Correct Answer: More mass is farther from axis.
Detailed Explanation: This tests mass distribution. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
31. JEE Advanced Exam-style Question: Derive rod I about centre using dm=(M/L)dx.
centre and end axesCorrect Answer: I=∫x²dm from -L/2 to L/2 = ML²/12.
Detailed Explanation: This tests integration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
32. JEE Advanced Exam-style Question: A ring and disc same M,R are coaxially joined. Find total I.
I = MR²Correct Answer: I=MR²+MR²/2=3MR²/2.
Detailed Explanation: This tests composite bodies. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
33. JEE Advanced Exam-style Question: Rod axis at distance L/4 from centre. Find I.
dCorrect Answer: I=ML²/12+M(L/4)²=7ML²/48.
Detailed Explanation: This tests shifted axes. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
34. JEE Advanced Exam-style Question: Disc has small concentric hole removed. How compute remaining I?
disc axesCorrect Answer: Subtract removed disc MOI from full disc MOI using same axis.
Detailed Explanation: This tests cut-out MOI. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
35. JEE Advanced Exam-style Question: For same M,R, compare rolling rotational KE at same omega.
solid vs hollowCorrect Answer: Hollow has larger I, so larger rotational KE.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
36. JEE Advanced Exam-style Question: If I=3MR²/2, find k.
dCorrect Answer: k=R sqrt(3/2).
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
37. JEE Advanced Exam-style Question: Why perpendicular axis theorem cannot apply to solid sphere?
Iz = Ix + IyCorrect Answer: It is only for plane lamina.
Detailed Explanation: This tests perpendicular theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
38. JEE Advanced Exam-style Question: Thick cylinder with R1=R, R2=2R. Find I.
Correct Answer: I=1/2 M(R²+4R²)=5MR²/2.
Detailed Explanation: This tests thick shell. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
39. JEE Advanced Exam-style Question: What axes are allowed in parallel axis theorem?
dCorrect Answer: Parallel axes, one through COM, separated by d.
Detailed Explanation: This tests parallel theorem condition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
40. JEE Advanced Exam-style Question: Why ring has larger I than disc?
I = MR²Correct Answer: More mass is farther from axis.
Detailed Explanation: This tests mass distribution. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
41. JEE Advanced Exam-style Question: Derive rod I about centre using dm=(M/L)dx.
centre and end axesCorrect Answer: I=∫x²dm from -L/2 to L/2 = ML²/12.
Detailed Explanation: This tests integration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
42. JEE Advanced Exam-style Question: A ring and disc same M,R are coaxially joined. Find total I.
I = MR²Correct Answer: I=MR²+MR²/2=3MR²/2.
Detailed Explanation: This tests composite bodies. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
43. JEE Advanced Exam-style Question: Rod axis at distance L/4 from centre. Find I.
dCorrect Answer: I=ML²/12+M(L/4)²=7ML²/48.
Detailed Explanation: This tests shifted axes. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
44. JEE Advanced Exam-style Question: Disc has small concentric hole removed. How compute remaining I?
disc axesCorrect Answer: Subtract removed disc MOI from full disc MOI using same axis.
Detailed Explanation: This tests cut-out MOI. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
45. JEE Advanced Exam-style Question: For same M,R, compare rolling rotational KE at same omega.
solid vs hollowCorrect Answer: Hollow has larger I, so larger rotational KE.
Detailed Explanation: This tests hollow vs solid. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
46. JEE Advanced Exam-style Question: If I=3MR²/2, find k.
dCorrect Answer: k=R sqrt(3/2).
Detailed Explanation: This tests radius of gyration. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
47. JEE Advanced Exam-style Question: Why perpendicular axis theorem cannot apply to solid sphere?
Iz = Ix + IyCorrect Answer: It is only for plane lamina.
Detailed Explanation: This tests perpendicular theorem. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
48. JEE Advanced Exam-style Question: Thick cylinder with R1=R, R2=2R. Find I.
Correct Answer: I=1/2 M(R²+4R²)=5MR²/2.
Detailed Explanation: This tests thick shell. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
49. JEE Advanced Exam-style Question: What axes are allowed in parallel axis theorem?
dCorrect Answer: Parallel axes, one through COM, separated by d.
Detailed Explanation: This tests parallel theorem condition. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
50. JEE Advanced Exam-style Question: Why ring has larger I than disc?
I = MR²Correct Answer: More mass is farther from axis.
Detailed Explanation: This tests mass distribution. Identify the body, axis, and whether an axis theorem is needed. Moment of inertia depends on mass distribution, not mass alone.
16

IB / IGCSE / A-Level Questions

Separate international banks with answers and explanations.

IB Questions

IB 1. Define moment of inertia.
Rotational inertia; sum/integral of mass times squared distance from axis.
IB 2. Point mass formula.
I=mr².
IB 3. Continuous body formula.
I=∫r²dm.
IB 4. Unit of I.
kg m².
IB 5. Dimension of I.
ML².
IB 6. Radius of gyration definition.
Distance k such that I=Mk².
IB 7. Formula for k.
k=sqrt(I/M).
IB 8. Parallel axis theorem.
I=Icm+Md².
IB 9. Perpendicular axis theorem.
Iz=Ix+Iy for plane lamina.
IB 10. Rod centre formula.
ML²/12.
IB 11. Rod end formula.
ML²/3.
IB 12. Ring central formula.
MR².
IB 13. Ring diameter formula.
MR²/2.
IB 14. Disc central formula.
MR²/2.
IB 15. Disc diameter formula.
MR²/4.
IB 16. Disc tangent formula.
3MR²/2.
IB 17. Solid cylinder formula.
MR²/2.
IB 18. Hollow cylinder formula.
MR².
IB 19. Thick cylinder formula.
1/2 M(R1²+R2²).
IB 20. Solid sphere formula.
2MR²/5.
IB 21. Hollow sphere formula.
2MR²/3.
IB 22. Why hollow larger?
Mass farther from axis.
IB 23. Can I be scalar?
For fixed axis, yes as a scalar value.
IB 24. MOI depends on axis?
Yes.
IB 25. Main exam tip.
Draw axis before choosing formula.

IGCSE Questions

IGCSE 1. What does moment of inertia describe?
Resistance to rotational motion.
IGCSE 2. Bigger mass farther from axis gives?
Larger moment of inertia.
IGCSE 3. Unit of moment of inertia.
kg m².
IGCSE 4. Ring or disc has larger I for same M,R?
Ring.
IGCSE 5. Solid or hollow sphere has larger I?
Hollow sphere.
IGCSE 6. Rod balanced at centre rotates about?
Centre axis.
IGCSE 7. Longer rod generally has larger?
Moment of inertia.
IGCSE 8. Radius of gyration symbol.
k.
IGCSE 9. Formula I=Mk² means?
Mass imagined at distance k.
IGCSE 10. Parallel axis adds what term?
Md².
IGCSE 11. Perpendicular axis applies to?
Plane lamina.
IGCSE 12. Disc central I.
MR²/2.
IGCSE 13. Ring central I.
MR².
IGCSE 14. Solid cylinder I.
MR²/2.
IGCSE 15. Hollow cylinder I.
MR².
IGCSE 16. Solid sphere I.
2MR²/5.
IGCSE 17. Hollow sphere I.
2MR²/3.
IGCSE 18. Axis selection important?
Yes.
IGCSE 19. Mass close to axis gives small?
Moment of inertia.
IGCSE 20. Flywheel stores energy due to?
Moment of inertia and rotation.
IGCSE 21. Rod end I compared to centre I.
Larger.
IGCSE 22. Disc diameter I compared to central I.
Half.
IGCSE 23. Tangent axis I is larger because?
Axis shifted from centre.
IGCSE 24. Composite body I is found by?
Adding parts about same axis.
IGCSE 25. Cut-out body I is found by?
Subtracting removed part.

A-Level Questions

A-Level 1. Derive I=∫r²dm.
Sum point masses and take continuous limit.
A-Level 2. Rod centre integration.
∫ from -L/2 to L/2 x²(M/L)dx=ML²/12.
A-Level 3. Rod end via integration.
∫0 to L x²(M/L)dx=ML²/3.
A-Level 4. Disc central formula.
MR²/2.
A-Level 5. Ring diameter using theorem.
MR²/2.
A-Level 6. Disc diameter using theorem.
MR²/4.
A-Level 7. Parallel axis theorem proof idea.
Expand (x+d)² and COM term vanishes.
A-Level 8. Perpendicular theorem condition.
Plane lamina only.
A-Level 9. Thick cylinder formula.
1/2M(R1²+R2²).
A-Level 10. Solid sphere formula.
2MR²/5.
A-Level 11. Spherical shell formula.
2MR²/3.
A-Level 12. Radius of gyration for rod centre.
L/sqrt12.
A-Level 13. Radius of gyration for ring.
R.
A-Level 14. Radius of gyration for disc central.
R/sqrt2.
A-Level 15. Composite I method.
Add or subtract each component about same axis.
A-Level 16. Cut-out I method.
Full body I minus removed part I after shifting axis if needed.
A-Level 17. Rolling energy relation.
K=1/2Mv²+1/2Iω².
A-Level 18. Hollow cylinder rolling has more rotational fraction because?
I is larger.
A-Level 19. Moment of inertia tensor idea.
MOI depends on axis direction in 3D.
A-Level 20. Axis through COM needed for which theorem?
Parallel axis theorem.
A-Level 21. Dimension of radius of gyration.
Length.
A-Level 22. Dimension of MOI.
ML².
A-Level 23. Why ring I>disc I?
Ring mass all at radius R.
A-Level 24. Why hollow sphere I>solid sphere I?
More mass farther from axis.
A-Level 25. Exam trap.
Do not use perpendicular theorem for 3D solid bodies.
17

Assertion Reason

30 assertion-reason questions.

1. Assertion: Moment of inertia depends on mass distribution. Reason: I contains r² term.
Answer: Both true; reason explains assertion.
Explanation: Verify the axis and body before judging the statement.
2. Assertion: Ring has larger I than disc for same M,R. Reason: Ring has more mass away from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
3. Assertion: Parallel axis theorem is I=Icm+Md². Reason: d is distance between parallel axes.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
4. Assertion: Perpendicular axis theorem applies to all 3D bodies. Reason: Iz=Ix+Iy.
Answer: Assertion false; theorem is for plane lamina.
Explanation: Verify the axis and body before judging the statement.
5. Assertion: Radius of gyration has dimension of length. Reason: I=Mk².
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
6. Assertion: Rod about end has smaller I than about centre. Reason: End axis is farther from COM.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
7. Assertion: Hollow sphere has larger I than solid sphere. Reason: Mass is farther from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
8. Assertion: Disc tangent I is 3MR²/2. Reason: Use parallel axis theorem on central axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
9. Assertion: Composite MOI can be added directly for different axes. Reason: MOI depends on axis.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
10. Assertion: Cut-out MOI uses subtraction. Reason: Removed part contributes negative inertia.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
11. Assertion: Moment of inertia depends on mass distribution. Reason: I contains r² term.
Answer: Both true; reason explains assertion.
Explanation: Verify the axis and body before judging the statement.
12. Assertion: Ring has larger I than disc for same M,R. Reason: Ring has more mass away from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
13. Assertion: Parallel axis theorem is I=Icm+Md². Reason: d is distance between parallel axes.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
14. Assertion: Perpendicular axis theorem applies to all 3D bodies. Reason: Iz=Ix+Iy.
Answer: Assertion false; theorem is for plane lamina.
Explanation: Verify the axis and body before judging the statement.
15. Assertion: Radius of gyration has dimension of length. Reason: I=Mk².
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
16. Assertion: Rod about end has smaller I than about centre. Reason: End axis is farther from COM.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
17. Assertion: Hollow sphere has larger I than solid sphere. Reason: Mass is farther from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
18. Assertion: Disc tangent I is 3MR²/2. Reason: Use parallel axis theorem on central axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
19. Assertion: Composite MOI can be added directly for different axes. Reason: MOI depends on axis.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
20. Assertion: Cut-out MOI uses subtraction. Reason: Removed part contributes negative inertia.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
21. Assertion: Moment of inertia depends on mass distribution. Reason: I contains r² term.
Answer: Both true; reason explains assertion.
Explanation: Verify the axis and body before judging the statement.
22. Assertion: Ring has larger I than disc for same M,R. Reason: Ring has more mass away from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
23. Assertion: Parallel axis theorem is I=Icm+Md². Reason: d is distance between parallel axes.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
24. Assertion: Perpendicular axis theorem applies to all 3D bodies. Reason: Iz=Ix+Iy.
Answer: Assertion false; theorem is for plane lamina.
Explanation: Verify the axis and body before judging the statement.
25. Assertion: Radius of gyration has dimension of length. Reason: I=Mk².
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
26. Assertion: Rod about end has smaller I than about centre. Reason: End axis is farther from COM.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
27. Assertion: Hollow sphere has larger I than solid sphere. Reason: Mass is farther from axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
28. Assertion: Disc tangent I is 3MR²/2. Reason: Use parallel axis theorem on central axis.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
29. Assertion: Composite MOI can be added directly for different axes. Reason: MOI depends on axis.
Answer: Assertion false; reason true.
Explanation: Verify the axis and body before judging the statement.
30. Assertion: Cut-out MOI uses subtraction. Reason: Removed part contributes negative inertia.
Answer: Both true.
Explanation: Verify the axis and body before judging the statement.
18

Case Study Questions

Case studies on rotating rod, flywheel, rolling bodies, comparison and axis theorem.

Case Study: Rotating rod
centre and end axesPassage: A rod rotates about centre and then about end. Questions: formula, axis shift, ratio, radius of gyration and energy implication.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: Centre I=ML²/12; end I=ML²/3 by parallel axis theorem.
Case Study: Flywheel
Passage: A flywheel has large mass near rim. Questions: why high I, energy storage, radius of gyration and angular speed effect.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: Mass far from axis gives large I and stores rotational energy.
Case Study: Solid and hollow cylinders rolling
Passage: Two cylinders with same M,R roll. Questions: compare I, rotational KE, acceleration and mass distribution.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: Hollow cylinder has larger I because mass is farther from axis.
Case Study: Disc and ring comparison
I = MR²Passage: A disc and ring of same M,R rotate about central axis. Questions: formulas, ratio, physical reason and k.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: Ring I=MR², disc I=MR²/2, so ring is twice.
Case Study: Sphere rolling
solid vs hollowPassage: Solid and hollow spheres roll with same M,R. Questions: compare I and rotational fraction.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: Hollow sphere has I=2MR²/3, solid has 2MR²/5.
Case Study: Parallel axis theorem
dPassage: A body axis is shifted parallel from COM. Questions: theorem, meaning of d and why I increases.
Questions: Identify body, axis, formula, comparison and theorem.
Answers and Detailed Explanation: I=Icm+Md², so shifted-axis I is larger.
19

Common Student Mistakes

Avoid these common errors before NEET/JEE numericals.

MR² vs 1/2MR²

Ring is MR²; disc and solid cylinder are MR²/2.

Wrong Axis

Formula changes when axis changes.

Forgetting Parallel Axis

Shifted axis needs +Md².

3D Perpendicular Theorem

Do not apply Iz=Ix+Iy to solid sphere or cylinder.

Sphere Confusion

Solid sphere is 2MR²/5; hollow sphere is 2MR²/3.

Radius of Gyration

k is equivalent distance, not always actual radius.

Diameter vs Central Axis

Disc diameter is MR²/4, central axis is MR²/2.

Searching for a Physics Tutor? If Moment of Inertia, Radius of Gyration, Axis Theorems or NEET/JEE numericals are not clear, contact Kumar Sir.

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