Searching for a Physics Tutor? If Potential Energy, Conservation of Energy or Mechanical Energy problems are not clear, contact Kumar Sir.
Phone: +91-9958461445 | Email: kumarsirphysics@gmail.com | Website: kumarphysicsclasses.com
U
CLASS 11 PHYSICS - WORK ENERGY POWER
Potential Energy and Conservation of Energy
Master gravitational potential energy, elastic potential energy, conservative forces, conservation of mechanical energy, numericals and PYQs.
CBSENEETJEE MainJEE AdvancedIBIGCSEA-Level
U
U = mghGravitational potential energy near Earth.
k
U = 1/2 kx²Elastic potential energy in a spring.
E
K + U = ConstantMechanical energy conservation.
∮
Wclosed = 0For conservative forces.
01
Potential Energy
Potential energy is stored energy due to position, shape or configuration.
Definition And Meaning
Potential energy exists because a force field or deformation can do work when the system changes configuration.
- It is scalar.
- SI unit is joule.
- Dimensions are [M L2 T-2].
- Its value depends on chosen reference level.
Stored Energy Idea
02
Gravitational Potential Energy
Near Earth, gravitational potential energy changes by mgh when a body is raised by height h.
Derivation
To lift a mass m slowly through height h, external force equals mg. Work done is stored as GPE.
W = Fs = mghU = mghReference level decides where U = 0.
Positive And Negative PE
Above reference level, U is usually positive. Below reference level, U can be negative. Only change in potential energy is physically important.
ΔU = mgΔhHeight Reference
Example: A 2 kg body is lifted by 5 m. Find gain in GPE. Take g=10.
Diagram: body moves vertically upward. Given m=2 kg, h=5 m. Formula ΔU=mgh. Calculation =2x10x5=100 J. Final answer: 100 J. Tip: height is vertical height, not path length.
Example: A 1 kg body falls 8 m. Find change in GPE. Take g=10.
ΔU=mg(hf-hi)=-mgh=-80 J. Final answer: -80 J. Common mistake: gravity does positive work, but GPE decreases.
03
Elastic Potential Energy
A stretched or compressed spring stores energy due to deformation.
Hooke's Law
F = kxk is spring constant and x is extension or compression from natural length.
Derivation
Spring force increases linearly from 0 to kx. Area under F-x graph is triangular.
U = ∫0x kx dx = 1/2kx²Spring System
Stretched Spring
Compressed Spring
F vs x Area
04
Conservative Force
A conservative force does path-independent work. Closed path work is zero.
Definition
Work done depends only on initial and final positions, not path.
Wclosed = 0Examples
- Gravitational force
- Spring force
- Electrostatic force
Path Independence
05
Non-Conservative Force
Non-conservative forces are path-dependent and often dissipate mechanical energy.
Examples And Meaning
- Friction
- Air resistance
- Viscous force
These forces convert mechanical energy into heat, sound or internal energy.
Comparison Table
| Conservative | Non-Conservative |
|---|---|
| Path independent | Path dependent |
| Closed path work zero | Closed path work non-zero |
| Stores recoverable energy | Dissipates mechanical energy |
| Gravity, spring, electrostatic | Friction, air resistance |
06
Conservation of Mechanical Energy
Very important: if only conservative forces act, total mechanical energy remains constant.
Derivation
Work by conservative force equals negative change in potential energy and also equals change in kinetic energy.
W = ΔK = -ΔUΔK + ΔU = 0K + U = ConstantK1 + U1 = K2 + U2
Energy Conversion
Need help with Conservation of Energy?
Contact Kumar Sir for one-to-one online Physics classes.
Phone: +91-9958461445Email: kumarsirphysics@gmail.comWebsite: kumarphysicsclasses.com
07
Applications
Ten premium application cards for conservation of mechanical energy.
Free Fall
Concept: GPE converts to KE.
mgh = 1/2mv²Exam tip: mass cancels.
Projectile Motion
Concept: energy depends on height and speed.
1/2mv² + mgh = constantExam tip: horizontal velocity stays unchanged without air drag.
Roller Coaster
Concept: height changes speed.
mgΔh = ΔKExam tip: use vertical height difference.
Pendulum
Concept: PE at extreme converts to KE at mean.
mgh = 1/2mv²Exam tip: tension does no work.
Spring Compression
Concept: compressed spring stores energy.
1/2kx²Exam tip: compression in metres.
Spring Extension
Concept: stretched spring also stores energy.
U = 1/2kx²Exam tip: energy same for ±x.
Vertical Circular Motion
Concept: height affects speed around loop.
K+U=constantExam tip: top is 2R above bottom.
Escape Velocity Concept
Concept: minimum energy to reach infinity.
K + U = 0Exam tip: total energy just zero.
Bungee Jump
Concept: GPE converts to spring PE.
mgh = 1/2kx²Exam tip: include extension distance carefully.
Daily Life
Concept: food, batteries, stretched bands and raised objects store energy.
Energy transforms, not disappears.Exam tip: mention losses if friction exists.
08
Important Graphs
SVG graphs for potential energy, spring energy and mechanical energy conversion.
Potential Energy vs Height
Linear graph; slope is mg.
Spring Force vs Extension
Area under F-x graph gives elastic PE.
Potential Energy vs Extension
Parabolic graph for spring PE.
Mechanical Energy
Total mechanical energy stays constant without losses.
Energy Conversion
As height decreases, PE converts into KE.
Losses Present
With non-conservative work, mechanical energy decreases.
09
High-Quality Numericals
Solved numerical bank with question, diagram, given, formula, calculation, final answer, exam tip and common mistake.
CBSE: A 2 kg body is raised by 4 m. Find gain in PE. Take g=10.
Diagram: vertical lift. Given m=2 kg, h=4 m. Formula U=mgh. Calculation 2x10x4=80 J. Final answer: 80 J. Exam tip: use vertical height. Common mistake: using path length.
NEET: A spring k=200 N/m is compressed 0.1 m. Find energy.
Diagram: compressed spring. U=1/2kx²=1/2x200x0.01=1 J. Final answer: 1 J. Tip: x in metres. Mistake: forgetting 1/2.
JEE Main: A body falls from height 20 m. Find speed at ground. g=10.
mgh=1/2mv². v=√(2gh)=√400=20 m/s. Tip: mass cancels. Mistake: using h as path only if vertical height.
JEE Advanced: Spring k=500 N/m compressed 0.2 m launches 2 kg block on smooth surface. Speed?
1/2kx²=1/2mv². 10=1v², so v=√10 m/s. Tip: no losses. Mistake: using kx instead of 1/2kx².
IB: A 1.5 kg ball is 6 m above reference. Find GPE. g=9.8.
U=mgh=1.5x9.8x6=88.2 J. Tip: state reference level. Mistake: PE absolute value depends on reference.
IGCSE: Weight 30 N lifted 2 m. Work stored as PE?
U=Wh=30x2=60 J. Tip: if weight is given, do not multiply by g again.
A-Level: Pendulum bob drops height 0.2 m. Find speed. g=9.8.
mgh=1/2mv², v=√(2x9.8x0.2)=1.98 m/s. Tip: tension does no work.
Roller coaster: Drops 15 m from rest. Find speed ignoring friction. g=10.
v=√(2gh)=√300=10√3 m/s. Tip: depends on height drop, not track shape.
10
NEET Question Bank
50 high-quality NEET-style MCQs on PE, spring energy, conservation and conservative forces.
1. NEET Exam-style Question: GPE near Earth is: A mv B mgh C 1/2mv² D kx
Answer: B. U=mgh.
2. Spring PE is: A kx B 1/2kx² C mgx D mv²
Answer: B.
3. Conservative force has closed path work: A positive B negative C zero D infinite
Answer: C.
4. Friction is: A conservative B non-conservative C always zero D electrostatic
Answer: B.
5. If height doubles, GPE: A half B same C double D four times
Answer: C.
6. If spring extension doubles, PE: A double B four times C half D same
Answer: B.
7. K+U is constant when: A friction acts B only conservative forces act C air drag acts D viscosity acts
Answer: B.
8. A 2 kg body at 5 m has GPE g=10: A 10 B 50 C 100 D 200 J
Answer: C.
9. Spring k=100, x=0.2 m. PE: A 1 B 2 C 4 D 20 J
Answer: B.
10. Gravity is: A conservative B non-conservative C viscous D frictional
Answer: A.
11. Spring force is: A conservative B non-conservative C dissipative D random
Answer: A.
12. Air resistance is: A conservative B non-conservative C gravitational D elastic
Answer: B.
13. GPE can be negative because: A mass negative B reference chosen above object C g zero D speed negative
Answer: B.
14. PE is a: A vector B scalar C tensor D direction
Answer: B.
15. Unit of PE is: A joule B watt C newton D pascal
Answer: A.
16. Dimension of PE: A MLT^-2 B ML²T^-2 C MLT^-1 D LT^-2
Answer: B.
17. Falling body loses: A KE B GPE C mass D charge
Answer: B.
18. Falling body gains: A KE B GPE C mass D spring energy
Answer: A.
19. Speed after falling height h: A √gh B √2gh C gh D 2gh
Answer: B.
20. F-x spring graph area is: A kx B 1/2kx² C mgx D zero
Answer: B.
21. Slope of spring F-x graph is: A x B k C 1/k D U
Answer: B.
22. U vs h graph is: A straight B parabola C circle D hyperbola
Answer: A.
23. Spring U vs x graph is: A straight B parabola C circle D constant
Answer: B.
24. Mechanical energy with friction: A conserved B decreases C always increases D zero
Answer: B generally, due to dissipation.
25. Pendulum at extreme has maximum: A KE B PE C speed D friction
Answer: B.
26. Pendulum at mean has maximum: A KE B PE C height D compression
Answer: A.
27. Work by conservative force equals: A ΔU B -ΔU C zero always D friction
Answer: B.
28. Work by gravity when body moves up height h: A mgh B -mgh C zero D 1/2kx²
Answer: B.
29. Change in GPE when body moves up h: A mgh B -mgh C zero D infinite
Answer: A.
30. If K decreases and no losses, U: A decreases B increases C same D negative only
Answer: B.
31. If U decreases by 40 J, K increases by: A 20 B 40 C 80 D 0 J
Answer: B if mechanical energy conserved.
32. Spring k=50, x=0.4. PE: A 2 B 4 C 8 D 16 J
Answer: B.
33. Height for 100 J GPE with m=2 kg, g=10: A 2 B 5 C 10 D 20 m
Answer: B.
34. Elastic PE depends on x sign? A yes B no, x² C only negative D only positive
Answer: B.
35. Electrostatic force is: A conservative B frictional C viscous D drag
Answer: A.
36. Viscous force is: A conservative B non-conservative C spring D gravity
Answer: B.
37. Reference level affects: A ΔU only B absolute U C mass D g
Answer: B; differences matter physically.
38. If only gravity acts, mechanical energy is: A conserved B lost C zero D infinite
Answer: A.
39. Bungee jump mainly converts GPE to: A thermal only B elastic PE and KE C mass D charge
Answer: B.
40. Roller coaster speed depends on: A track length only B vertical height drop C color D mass only
Answer: B ignoring losses.
41. If non-conservative work Wnc acts, equation is: A K+U const B ΔK+ΔU=Wnc C U=0 D K=0
Answer: B.
42. Friction work is usually: A positive B negative C zero D conservative
Answer: B.
43. Spring stores 8 J at x. At 2x stores: A 8 B 16 C 32 D 64 J
Answer: C.
44. Body of weight 20 N raised 3 m gains PE: A 20 B 40 C 60 D 80 J
Answer: C.
45. Work by gravity in closed path: A zero B positive C negative D infinite
Answer: A.
46. Work by friction in closed path: A zero always B non-zero generally C conservative D mgh
Answer: B.
47. Mechanical energy includes: A K only B U only C K+U D force only
Answer: C.
48. At maximum height of vertical throw, KE is: A maximum B minimum C infinite D same as ground
Answer: B, zero at top if vertical velocity becomes zero.
49. Potential energy exists due to: A position/configuration B speed only C time only D color
Answer: A.
50. Conservation of energy fails mechanically when: A no friction B only gravity C friction/drag acts D spring ideal
Answer: C; total energy still conserved including heat.
11
JEE Main Question Bank
50 difficult JEE Main questions.
1. A 2 kg body rises 10 m. ΔU, g=10?
ΔU=mgh=200 J.
2. Spring k=400, x=0.05. U?
U=1/2x400x0.0025=0.5 J.
3. Drop height 45 m. Speed, g=10?
v=√900=30 m/s.
4. Pendulum bob drops 0.8 m. Speed, g=10?
v=√16=4 m/s.
5. Spring energy 9 J, k=200. Compression?
x=√(2U/k)=√(18/200)=0.3 m.
6. Body falls 20 m with friction loss 50 J, m=5 kg, g=10. Final KE?
mgh=1000 J. K=950 J.
7. Roller coaster drops 5 m then rises 2 m. Speed from rest at final point g=10?
Net drop=3 m. v=√60 m/s.
8. Projectile launched speed u. Speed at height h?
1/2mu²=1/2mv²+mgh, so v=√(u²-2gh).
9. Spring k, compression x launches mass m. Speed?
v=x√(k/m).
10. Same spring with friction μmg over distance x.
1/2mv²=1/2kx²-μmgx.
11. Work by gravity for downward height h.
+mgh.
12. Change in PE for downward height h.
-mgh.
13. If U=mgh, slope of U-h graph?
mg.
14. If U=1/2kx², slope of U-x graph?
dU/dx=kx.
15. Conservative force from U(x): F?
F=-dU/dx.
16. U=3x². Force?
F=-6x.
17. U=5x. Force?
F=-5 N.
18. Work by conservative force from U1=40 J to U2=10 J.
W=-(U2-U1)=30 J.
19. K1=20,U1=80,K2=50. Find U2.
Total=100, U2=50 J.
20. K+U=200, K=75. U?
125 J.
21. A body with speed 10 m/s rises vertically. Max height g=10?
1/2mv²=mgh, h=5 m.
22. Speed at height 3 m for upward throw u=10, g=10?
v=√(100-60)=√40=2√10.
23. Spring k=100, x=0.4 launches m=2. Speed?
U=8 J. 1/2x2v²=8, v=2√2.
24. Friction work -30 J, ΔK+ΔU?
Equals Wnc=-30 J.
25. Closed path work for spring force?
Zero.
26. Closed path work for friction?
Negative non-zero generally.
27. A 1 kg block slides from height 5 m, loses 10 J. Final speed g=10?
K=50-10=40 J, v=√80=4√5.
28. Pendulum length L released from horizontal. Speed at bottom?
Drop=L, v=√(2gL).
29. Loop top height 2R above bottom. Minimum bottom speed for reaching top by energy only?
v=2√(gR).
30. Complete loop bottom speed condition?
v_bottom ≥ √(5gR).
31. Spring U at x and 2x ratio?
1:4.
32. GPE at h and 2h ratio?
1:2.
33. Mass doubled, same height. GPE?
Doubles.
34. k doubled, same x. Spring PE?
Doubles.
35. x tripled, spring PE?
Nine times.
36. Bungee: mgh=1/2kx². Solve k.
k=2mgh/x².
37. A conservative force has potential U=x³. Work 1 to 2?
W=-(8-1)=-7 J.
38. A particle moves from U=100 to U=40 with no losses. ΔK?
+60 J.
39. U increases by 25 J with no losses. ΔK?
-25 J.
40. Mechanical energy initially 150 J and Wnc=-20 J. Final mechanical energy?
130 J.
41. Spring compressed 0.2 m stores 4 J. k?
k=2U/x²=8/0.04=200 N/m.
42. A 4 kg mass has GPE 200 J. Height g=10?
h=200/(40)=5 m.
43. Weight 50 N has GPE 300 J. Height?
h=6 m.
44. Free fall speed after losing PE 500 J for mass 10 kg.
1/2mv²=500, v=10 m/s.
45. Projectile speed at same height as launch ignoring air.
Same magnitude as launch speed.
46. At highest point of projectile, mechanical energy?
Same as initial, but KE is due to horizontal velocity only.
47. If reference level shifted upward by H, PE changes by?
All U values decrease by mgH, differences unchanged.
48. Force from spring potential U=1/2kx²?
F=-dU/dx=-kx.
49. Energy lost to friction over rough horizontal s?
μmgs.
50. Equation with non-conservative work?
K1+U1+Wnc=K2+U2.
12
JEE Advanced Question Bank
50 difficult JEE Advanced questions with complete compact solutions.
1. U=ax²+bx. Find force.
F=-dU/dx=-(2ax+b).
2. U=kx⁴/4. Force?
F=-kx³.
3. U=A/x. Force?
F=-dU/dx=A/x².
4. Work by conservative force from x1 to x2 if U=x².
W=-(x2²-x1²)=x1²-x2².
5. Spring from x=A to x=0. Work by spring?
W=1/2kA².
6. Spring from x=A to x=2A. Work by spring?
W=-[1/2k(4A²-A²)]=-3kA²/2.
7. Block released from spring compression A on rough surface. Distance until stop after leaving spring?
Energy available 1/2kA² is dissipated by friction μmg s, so total stop distance s=kA²/(2μmg) if friction acts throughout.
8. Pendulum released from angle θ. Speed at bottom?
Drop h=L(1-cosθ). v=√(2gL(1-cosθ)).
9. Roller coaster drop h with friction work -fL. Speed?
1/2mv²=mgh-fL.
10. Projectile launched speed u at angle. Speed at height h?
v²=u²-2gh, independent of launch angle if height reached.
11. Escape condition at planet surface using U=-GMm/R.
1/2mv²-GMm/R=0, so v=√(2GM/R).
12. Force F=-kx has potential?
U=1/2kx² + constant.
13. Force F=-a/x² has potential?
Since F=-dU/dx, dU/dx=a/x², U=-a/x+C.
14. Stable equilibrium condition in U-x graph?
dU/dx=0 and d²U/dx²>0.
15. Unstable equilibrium condition?
dU/dx=0 and d²U/dx²<0.
16. Neutral equilibrium condition?
Potential energy locally constant; force zero over region.
17. U=x⁴-2x² equilibrium points.
dU/dx=4x³-4x=4x(x²-1)=0, so x=0,±1.
18. Classify U=x⁴-2x² at x=0 and ±1.
d²U=12x²-4. At 0 negative unstable; at ±1 positive stable.
19. If total energy E and U(x)>E region?
Classically forbidden because K=E-U would be negative.
20. Turning point in U graph?
Where K=0, so E=U.
21. Vertical circle minimum bottom speed for string.
v_bottom=√(5gR).
22. Speed at top for just complete loop.
v_top=√(gR).
23. Bottom to top energy relation in loop.
1/2mv_b²=1/2mv_t²+mg(2R).
24. Spring and gravity vertical: mass pulled down x from natural length. Energy?
Use K+1/2kx²-mgx=constant with chosen reference.
25. Extension at equilibrium vertical spring.
kx=mg, so x=mg/k.
26. Work by gravity over closed path.
Zero.
27. Work by friction over closed path length L.
-μmgL on horizontal rough path.
28. Conservative force field test concept.
Closed-loop work zero or curl zero in suitable region.
29. U=mgy. Force vector?
F=-∇U=-mg j.
30. U=1/2k(x²+y²). Force?
F=-kx i - ky j.
31. If Wnc=-Q, mechanical energy changes by?
Δ(K+U)=Wnc=-Q.
32. Bungee maximum extension equation with rope natural length L and fall distance L+x.
mg(L+x)=1/2kx² at lowest point if starting from rest and no losses.
33. Two springs k1,k2 in parallel equivalent energy.
U=1/2(k1+k2)x², so k_eq=k1+k2.
34. Two springs in series equivalent constant.
1/k_eq=1/k1+1/k2.
35. Energy stored in series springs under force F.
U=F²/(2k_eq).
36. Energy stored in spring if force at extension x is F.
U=1/2Fx because graph is triangle.
37. Projectile at same height speed relation with air absent.
Speed magnitude is same at same height by energy conservation.
38. A bead slides on smooth track from height h. Speed at bottom.
v=√2gh independent of track shape.
39. Same bead with friction work Wf.
1/2mv²=mgh+Wf, where Wf negative.
40. Particle in potential U=a/x². Force?
F=-dU/dx=2a/x³.
41. Potential energy minimum means force?
Force is zero at minimum, and equilibrium is stable.
42. If total energy line touches U curve.
That point is turning point or equilibrium depending on slope.
43. Work by spring over x=A to x=-A.
U initial = U final, so work by spring is zero.
44. Max speed in spring oscillation amplitude A.
1/2kA²=1/2mv², v=A√(k/m).
45. Height reached by spring launch vertical.
1/2kx²=mgh if all spring energy becomes GPE.
46. Include initial KE in energy equation.
K1+U1+Wnc=K2+U2.
47. Potential U=ax. Motion direction if a positive.
F=-a, so force toward negative x.
48. Potential U=-ax. Force?
F=+a, toward positive x.
49. Work by conservative force around rectangle.
Zero, if the force is conservative over the region.
50. Mechanical energy graph with friction should be:
Decreasing with distance/time because Wnc is negative.
13
IB / IGCSE / A-Level Questions
Separate international question sets with answers and explanations.
IB Questions - 25
IB 1. Define potential energy.
Stored energy due to position or configuration.
IB 2. Formula for GPE near Earth?
U=mgh.
IB 3. Formula for elastic PE?
U=1/2kx².
IB 4. State conservation of mechanical energy.
K+U remains constant if only conservative forces act.
IB 5. Work by conservative force in closed path?
Zero.
IB 6. Name a non-conservative force.
Friction or air resistance.
IB 7. GPE of 2 kg at 3 m, g=9.8.
58.8 J.
IB 8. Spring k=100, x=0.1. Energy?
0.5 J.
IB 9. Why reference level matters?
Absolute PE depends on zero level; changes are physical.
IB 10. Does friction conserve mechanical energy?
No, it dissipates mechanical energy.
IB 11. Falling from rest height h speed?
√2gh.
IB 12. Pendulum energy at lowest point?
Kinetic energy maximum.
IB 13. PE at highest pendulum position?
Maximum relative to lowest point.
IB 14. Conservative examples?
Gravity, spring and electrostatic forces.
IB 15. Work by gravity equals what PE change?
Wg=-ΔU.
IB 16. Mechanical energy with air drag?
Not conserved mechanically.
IB 17. U-h graph shape?
Straight line.
IB 18. U-x spring graph shape?
Parabola.
IB 19. F-x spring graph area?
Elastic potential energy.
IB 20. Unit of PE?
Joule.
IB 21. Dimension of PE?
ML²T^-2.
IB 22. Roller coaster speed depends on?
Vertical height change, ignoring losses.
IB 23. Bungee jump energy conversion?
GPE to KE and elastic PE.
IB 24. If U decreases by 30 J, K changes by?
K increases by 30 J if no losses.
IB 25. Why PE is scalar?
It has magnitude/sign relative to reference, no direction.
IGCSE Questions - 25
IGCSE 1. What is gravitational potential energy?
Energy stored due to height in a gravitational field.
IGCSE 2. Formula for GPE?
GPE=mgh or weight x height.
IGCSE 3. Unit of energy?
Joule.
IGCSE 4. Weight 20 N raised 5 m. GPE?
100 J.
IGCSE 5. Mass 2 kg raised 3 m, g=10. GPE?
60 J.
IGCSE 6. A falling object loses what energy?
Gravitational potential energy.
IGCSE 7. A falling object gains what energy?
Kinetic energy.
IGCSE 8. Spring stores which energy?
Elastic potential energy.
IGCSE 9. Friction changes mechanical energy into?
Heat/internal energy.
IGCSE 10. Energy is conserved means?
Total energy is not created or destroyed.
IGCSE 11. GPE depends on height or speed?
Height.
IGCSE 12. KE depends on height or speed?
Speed.
IGCSE 13. Raised book has what energy?
Gravitational potential energy.
IGCSE 14. Stretched rubber band has?
Elastic potential energy.
IGCSE 15. Weight 50 N lifted 2 m.
100 J.
IGCSE 16. What happens to GPE when height doubles?
It doubles.
IGCSE 17. What happens to spring energy when extension doubles?
It becomes four times.
IGCSE 18. Is friction useful in real life?
Yes, but it dissipates mechanical energy.
IGCSE 19. Roller coaster at top has high?
Potential energy.
IGCSE 20. Roller coaster at bottom has high?
Kinetic energy.
IGCSE 21. Unit of height in mgh?
Metre.
IGCSE 22. Unit of mass in mgh?
Kilogram.
IGCSE 23. If no energy lost, GPE lost equals?
KE gained.
IGCSE 24. Does path length matter for GPE?
No, vertical height change matters.
IGCSE 25. Name an energy store in a compressed spring.
Elastic potential energy store.
A-Level Questions - 25
A-Level 1. Define conservative force.
A force whose work is path independent and closed-loop work is zero.
A-Level 2. Relationship between force and potential in 1D.
F=-dU/dx.
A-Level 3. U=4x². Find F.
F=-8x.
A-Level 4. Spring energy from x1 to x2.
ΔU=1/2k(x2²-x1²).
A-Level 5. Work by spring from x1 to x2.
W=-1/2k(x2²-x1²).
A-Level 6. Mechanical energy with Wnc.
K1+U1+Wnc=K2+U2.
A-Level 7. Pendulum drop h speed.
v=√2gh.
A-Level 8. Projectile speed at height h.
v²=u²-2gh.
A-Level 9. Friction work on horizontal distance s.
W=-μmgs.
A-Level 10. Energy stored in spring under final force F and extension x.
U=1/2Fx.
A-Level 11. U=-GMm/r, escape speed.
v=√(2GM/r).
A-Level 12. Condition for stable equilibrium.
Potential energy minimum.
A-Level 13. Condition for turning point.
K=0 and E=U.
A-Level 14. U=mgy force.
F=-mg j.
A-Level 15. Work by conservative force from U=7 to U=2.
W=5 J.
A-Level 16. Work by conservative force if U increases by 12 J.
W=-12 J.
A-Level 17. Spring k=80, x=0.5.
U=10 J.
A-Level 18. GPE of 0.5 kg at 4 m, g=9.8.
19.6 J.
A-Level 19. Vertical circle energy bottom to top.
1/2mv_b²=1/2mv_t²+2mgR.
A-Level 20. Why normal does no work on fixed smooth track?
Normal is perpendicular to displacement.
A-Level 21. Does tension do work in ideal pendulum?
No, tension is radial and displacement tangential.
A-Level 22. Energy lost to drag appears as?
Thermal/internal energy of air and object.
A-Level 23. Work by gravity around closed loop.
Zero.
A-Level 24. U=x⁴ has equilibrium at?
x=0, stable minimum.
A-Level 25. U=-x² equilibrium at x=0?
Unstable maximum.
14
Assertion Reason
30 assertion-reason questions with hidden answers.
1. Assertion: Potential energy is scalar. Reason: It has no direction.
Both true; reason explains assertion.
2. Assertion: GPE depends on reference level. Reason: Only PE differences are physically important.
Both true.
3. Assertion: Spring PE is proportional to x². Reason: Spring force is proportional to x.
Both true; integration gives x² dependence.
4. Assertion: Gravity is conservative. Reason: Work by gravity in closed path is zero.
Both true.
5. Assertion: Friction is conservative. Reason: Friction dissipates energy.
Assertion false, reason true.
6. Assertion: Mechanical energy is always conserved. Reason: Total energy is conserved.
Assertion false; mechanical energy is not conserved with non-conservative forces.
7. Assertion: K+U is constant with only conservative forces. Reason: ΔK+ΔU=0.
Both true.
8. Assertion: Work by conservative force is -ΔU. Reason: Potential energy is defined for conservative forces.
Both true.
9. Assertion: Elastic PE is same for +x and -x. Reason: U=1/2kx².
Both true.
10. Assertion: Falling body loses GPE. Reason: Height decreases.
Both true.
11. Assertion: Falling body gains KE if no losses. Reason: Mechanical energy is conserved.
Both true.
12. Assertion: PE can be negative. Reason: Reference can be chosen above object.
Both true.
13. Assertion: U=mgh depends on vertical height. Reason: Gravity near Earth is uniform.
Both true.
14. Assertion: Air resistance conserves mechanical energy. Reason: Air resistance is non-conservative.
Assertion false, reason true.
15. Assertion: Closed path work by spring is zero. Reason: Spring force is conservative.
Both true.
16. Assertion: Work by friction in closed path is zero. Reason: Friction is conservative.
Both false generally.
17. Assertion: Pendulum tension does no work. Reason: Tension is perpendicular to displacement.
Both true for ideal pendulum.
18. Assertion: Normal does no work on fixed smooth track. Reason: It is perpendicular to displacement.
Both true.
19. Assertion: PE vs height graph is straight. Reason: U=mgh.
Both true.
20. Assertion: Spring U-x graph is parabolic. Reason: U=1/2kx².
Both true.
21. Assertion: Spring F-x graph area gives PE. Reason: Work stored equals ∫Fdx.
Both true.
22. Assertion: Escape velocity total energy is zero. Reason: At infinity U and K are zero for minimum escape.
Both true.
23. Assertion: With friction, K1+U1=K2+U2. Reason: Friction is non-conservative.
Assertion false, reason true.
24. Assertion: Non-conservative work changes mechanical energy. Reason: Δ(K+U)=Wnc.
Both true.
25. Assertion: A roller coaster is faster at lower height. Reason: PE converts to KE.
Both true ignoring losses.
26. Assertion: Projectile speed same at same height without air drag. Reason: Mechanical energy conserved.
Both true.
27. Assertion: Spring constant is slope of F-x graph. Reason: F=kx.
Both true.
28. Assertion: Conservative force can store potential energy. Reason: Work is recoverable.
Both true.
29. Assertion: PE is frame/reference independent absolutely. Reason: zero level is arbitrary.
Assertion false, reason true.
30. Assertion: Energy conservation can solve without time. Reason: It directly relates positions and speeds.
Both true.
15
Case Study Questions
Case studies on falling body, spring system, roller coaster, pendulum and projectile motion.
Case 1: Falling body from height 20 m. Find speed just before ground, g=10.
GPE lost becomes KE: mgh=1/2mv². v=√(2x10x20)=20 m/s. Explanation: mass cancels.
Case 2: Spring k=300 N/m compressed 0.2 m launches 1.5 kg block.
Spring energy=1/2kx²=6 J. 1/2mv²=6, so v=√8=2√2 m/s. Explanation: compression energy converts to KE.
Case 3: Roller coaster drops 12 m and loses 30 J per kg to friction. Find v, g=10.
Energy per kg: gh-loss=120-30=90 J/kg. 1/2v²=90, v=√180=6√5 m/s.
Case 4: Pendulum released from height 0.45 m above lowest point.
mgh=1/2mv². v=√(2x10x0.45)=3 m/s. Tension does no work.
Case 5: Projectile launched at 30 m/s. Find speed at height 20 m, g=10.
v²=u²-2gh=900-400=500. v=10√5 m/s. Explanation: energy depends on height only, not path.
16
Common Student Mistakes
Avoid these errors in potential energy and conservation problems.
Confusing PE And KE
PE depends on position/configuration. KE depends on speed.
Wrong Reference Level
Absolute GPE changes with reference; energy differences matter.
Sign Errors In PE
Going up increases GPE; falling decreases GPE.
Forgetting Spring Energy
Use 1/2kx², not kx or kx².
Misusing Conservation
K+U is constant only when non-conservative work is zero.
Ignoring Non-Conservative Work
Use K1+U1+Wnc=K2+U2 when friction or drag is present.
Searching for a Physics Tutor? If Potential Energy, Conservation of Energy or Mechanical Energy problems are not clear, contact Kumar Sir.
Phone: +91-9958461445 | Email: kumarsirphysics@gmail.com | Website: kumarphysicsclasses.com