Gravitational Potential and Potential Energy

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Gravitational Potential and Potential Energy

Master gravitational potential, potential energy, equipotential surfaces, escape energy, numericals and PYQs.

CBSENEETJEE MainJEE AdvancedIBIGCSEA-Level

Gravitational Potential

Gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point without changing kinetic energy. It is a scalar quantity, so potentials add algebraically.

V = W/m
Unit: J kg-1.
Dimensions: [L2T-2].
Nature: Scalar.
Meaning: Energy per unit mass.

Work Per Unit Mass

Mass Mtest mass

Scalar Potential

V = V1 + V2

Potential Level

potential well

Gravitational Potential Energy

Gravitational potential energy is the energy stored due to the position of two masses in a gravitational field. Taking potential energy zero at infinity, the potential energy of two masses separated by r is negative because gravity is attractive.

U = -GMm/r

Derivation Idea

Work done by an external agent in slowly bringing mass m from infinity to distance r against gravity becomes stored potential energy. Since gravity pulls inward, the final potential energy relative to infinity is negative.

U(r) - U(infinity) = -GMm/r

Negative Energy Diagram

rU is negative for bound system

Potential Due to Point Mass

For a point mass M, gravitational potential at distance r is obtained by dividing potential energy by test mass m.

V = -GM/r

The value is negative when infinity is chosen as zero. As r increases, V approaches zero from the negative side.

Point Mass Potential

rMpoint

V vs r

r0V negative

Exam Tip

Potential is scalar. Do not attach direction to V. Direction belongs to gravitational field, not potential.

Potential Due to System of Masses

Potential follows superposition. Find potential due to each mass at the point and add algebraically.

V = sum(-GMi/ri)

Two-Mass System

r1r2

V = -GM1/r1 - GM2/r2.

Three-Mass System

Add all three scalar potentials.

Solved Example

Two equal masses M are each at distance r from point P. Find V at P.

Show Answer
V = -GM/r - GM/r = -2GM/r. Exam tip: no vector resolution is needed for potential.

Potential Due to Sphere

A uniform sphere behaves as a point mass for external points. Inside the sphere, potential changes smoothly and remains finite at the centre.

Outside Sphere

V = -GM/r

Surface

V = -GM/R

R

Inside Sphere

V = -(GM/2R3)(3R2 - r2)

r

Derivation Snapshot

Outside the sphere, shell theorem gives point-mass behavior. Inside, the field changes linearly with r for a uniform sphere, and integration of field gives the inside potential formula.

Sphere Potential Graph

RVr

Equipotential Surfaces

An equipotential surface is a surface on which gravitational potential is the same at every point. No work is done in moving a mass along an equipotential surface.

Equipotential Circles

same V on each circle

Field Lines

Field lines are perpendicular to equipotential surfaces.

No Work

If V is same at two points, potential energy change is zero: delta U = m delta V = 0.

Escape Energy Concept

To escape from Earth's gravitational field, an object must be supplied enough energy to raise its total mechanical energy to zero at infinity.

ve = sqrt(2GM/R)
Escape energy = GMm/R

Energy Interpretation

At surface, U = -GMm/R. To make total energy zero at infinity, required kinetic energy is GMm/R.

Escape Diagram

escape path

Mass Independence

Escape velocity is independent of projectile mass because both required energy and kinetic energy contain m, which cancels.

Important Graphs

Potential vs Distance

r0

V approaches zero from negative side.

Potential Energy vs Distance

rU

Same shape as potential, scaled by m.

Inside Sphere Potential

Rr

Parabolic inside a uniform sphere.

Outside Sphere Potential

r

Follows -GM/r.

Equipotential Representation

Concentric surfaces around point mass.

Important Formula Table

ConceptFormulaMeaningExam Tip
PotentialV = W/mWork per unit mass.Scalar quantity.
Potential EnergyU = -GMm/rEnergy of two-mass system.Negative for bound pair.
Point Mass PotentialV = -GM/rPotential at distance r.Zero at infinity.
System of MassesV = sum(-GMi/ri)Scalar superposition.No vector components.
Sphere OutsideV = -GM/rExternal point.Use centre distance.
Sphere SurfaceV = -GM/RAt r=R.Matches outside formula.
Sphere InsideV = -(GM/2R3)(3R2-r2)Uniform sphere.Finite at centre.
Escape Velocityve = sqrt(2GM/R)Minimum launch speed.Independent of mass.
Escape EnergyE = GMm/REnergy needed from surface.Equals -U surface in magnitude.

Applications

Satellites

Concept: Satellite energy uses potential energy plus kinetic energy.

Formula: U=-GMm/r.

Exam Tip: Use orbital radius from centre.

Escape Velocity

Concept: Escape means total energy reaches zero.

Formula: ve=sqrt(2GM/R).

Exam Tip: Independent of projectile mass.

Space Missions

Concept: Energy planning uses potential changes between orbits.

Formula: delta U = GMm(1/r1-1/r2).

Exam Tip: Track signs carefully.

Orbital Motion

Concept: Circular orbit has total energy negative.

Formula: E=-GMm/(2r).

Exam Tip: Bound orbit means E<0.

Planetary Systems

Concept: Potential wells explain bound planets.

Formula: V=-GM/r.

Exam Tip: Larger M means deeper well.

Potential Wells

Concept: Negative potential indicates binding.

Formula: Escape needs energy equal to well depth.

Exam Tip: Zero level is infinity.

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High-Quality Solved Numericals

CBSE Numerical

Question: Find potential due to Earth at surface.

Show Answer

Given: radius R. Formula: V=-GM/R. Calculation: Substitute Earth M and R if values are given. Final Answer: -GM/R. Exam Tip: Potential is per unit mass. Common Mistake: Multiplying by test mass.

NEET Numerical

Question: At distance 2R from Earth's centre, find potential in terms of surface potential Vs.

Show Answer

V=-GM/(2R)=Vs/2 because Vs = -GM/R. Since both are negative, V becomes less negative.

JEE Main Numerical

Question: Two equal masses M are at distance r from P. Find potential at P.

Show Answer

V=-GM/r-GM/r=-2GM/r. Potential is scalar, so add signs directly.

JEE Advanced Numerical

Question: Find escape energy for mass m from Earth's surface.

Show Answer

Initial U=-GMm/R, final energy at infinity is zero. Required energy = GMm/R. This is independent of path.

IB Numerical

Question: If V=-20 J kg-1, find potential energy of 3 kg mass.

Show Answer
U=mV=3(-20)=-60 J.

IGCSE Numerical

Question: If 200 J work is done per 5 kg, find potential difference.

Show Answer
Delta V=W/m=200/5=40 J kg-1.

A-Level Numerical

Question: At centre of uniform sphere, find V.

Show Answer
Put r=0 in inside formula: V=-(GM/2R3)(3R2)=-3GM/(2R).

NEET Question Bank

NEET Question 1

Gravitational potential is defined as?

Show Answer
Work done per unit mass in bringing mass from infinity.

NEET Question 2

Unit of V?

Show Answer
J kg-1.

NEET Question 3

Dimension of V?

Show Answer
[L2T-2].

NEET Question 4

Potential due to point mass?

Show Answer
V=-GM/r.

NEET Question 5

Potential energy of two masses?

Show Answer
U=-GMm/r.

NEET Question 6

Potential is scalar or vector?

Show Answer
Scalar.

NEET Question 7

Potential at infinity is usually?

Show Answer
Zero.

NEET Question 8

Why is gravitational potential negative?

Show Answer
Zero is at infinity and gravity is attractive.

NEET Question 9

Potential on equipotential surface is?

Show Answer
Same everywhere.

NEET Question 10

Work done along equipotential surface?

Show Answer
Zero.

NEET Question 11

Potential due to many masses is found by?

Show Answer
Scalar superposition.

NEET Question 12

Outside sphere potential?

Show Answer
-GM/r.

NEET Question 13

Surface potential of sphere?

Show Answer
-GM/R.

NEET Question 14

Escape velocity formula?

Show Answer
sqrt(2GM/R).

NEET Question 15

Escape energy from surface?

Show Answer
GMm/R.

NEET Question 16

Escape velocity depends on projectile mass?

Show Answer
No.

NEET Question 17

Potential energy equals mass times?

Show Answer
Potential, U=mV.

NEET Question 18

At r increases, V becomes?

Show Answer
Less negative, approaches zero.

NEET Question 19

V at r=2R compared to surface?

Show Answer
Half of surface value.

NEET Question 20

Potential inside sphere is finite at centre?

Show Answer
Yes.

NEET Question 21

V at centre of uniform sphere?

Show Answer
-3GM/(2R).

NEET Question 22

Potential graph for point mass lies below?

Show Answer
Zero axis.

NEET Question 23

Equipotential surfaces around point mass are?

Show Answer
Concentric spheres.

NEET Question 24

Field lines meet equipotential surface at?

Show Answer
Right angle.

NEET Question 25

V due to two equal masses each at r?

Show Answer
-2GM/r.

NEET Question 26

If V=-10 J/kg and m=2 kg, U?

Show Answer
-20 J.

NEET Question 27

Is potential direction-dependent?

Show Answer
No.

NEET Question 28

Potential energy of bound satellite is?

Show Answer
Negative.

NEET Question 29

Total energy in circular orbit?

Show Answer
-GMm/(2r).

NEET Question 30

Potential energy in circular orbit?

Show Answer
-GMm/r.

NEET Question 31

Kinetic energy in circular orbit?

Show Answer
GMm/(2r).

NEET Question 32

Potential difference unit?

Show Answer
J kg-1.

NEET Question 33

Negative sign in U indicates?

Show Answer
Attractive bound state.

NEET Question 34

Escape condition total energy?

Show Answer
Zero or positive at infinity.

NEET Question 35

At surface, energy needed to escape is?

Show Answer
Equal to magnitude of potential energy.

NEET Question 36

If mass doubles, escape velocity?

Show Answer
Unchanged for projectile mass; changes if planet mass doubles.

NEET Question 37

If planet mass doubles, escape velocity?

Show Answer
sqrt2 times.

NEET Question 38

If planet radius doubles, escape velocity?

Show Answer
1/sqrt2 times for same mass.

NEET Question 39

Sphere inside potential is maximum or minimum at centre?

Show Answer
Most negative at centre.

NEET Question 40

Potential is continuous at sphere surface?

Show Answer
Yes.

NEET Question 41

Can potential be zero at finite r for one mass?

Show Answer
No, with infinity reference it approaches zero at infinity.

NEET Question 42

Potential due to equal opposite directions cancels like vectors?

Show Answer
No, potential is scalar.

NEET Question 43

V due to two equal masses on opposite sides at same r?

Show Answer
-2GM/r, not zero.

NEET Question 44

Potential well depth increases with?

Show Answer
Mass M.

NEET Question 45

Potential energy change delta U equals?

Show Answer
m delta V.

NEET Question 46

Work by external agent slowly equals?

Show Answer
Change in potential energy.

NEET Question 47

Work by gravity equals?

Show Answer
-delta U.

NEET Question 48

Potential inside shell is?

Show Answer
Constant, equal to surface potential.

NEET Question 49

Field on equipotential surface tangential direction?

Show Answer
No tangential component.

NEET Question 50

Main sign mistake in potential?

Show Answer
Forgetting negative sign with infinity reference.

JEE Main Question Bank

JEE Main 1

Find V at distance 3R in terms of surface V.

Show Answer
V=Vs/3.

JEE Main 2

Find U if V=-5 and m=4.

Show Answer
U=-20 J.

JEE Main 3

Two masses M,2M at same r. Potential?

Show Answer
-3GM/r.

JEE Main 4

Three equal masses at distance a from P. Potential?

Show Answer
-3GM/a.

JEE Main 5

At sphere centre potential?

Show Answer
-3GM/(2R).

JEE Main 6

At sphere surface potential?

Show Answer
-GM/R.

JEE Main 7

At r=R/2 inside sphere potential?

Show Answer
-(GM/2R3)(3R2-R2/4)=-11GM/(8R).

JEE Main 8

Escape energy at radius r?

Show Answer
GMm/r.

JEE Main 9

Escape speed at radius r?

Show Answer
sqrt(2GM/r).

JEE Main 10

V due to equal masses at square corners at centre?

Show Answer
Each distance a/sqrt2, so V=-4GM/(a/sqrt2)=-4sqrt2 GM/a.

JEE Main 11

V at centre of equilateral triangle side a due to three masses M?

Show Answer
Distance a/sqrt3, so V=-3sqrt3 GM/a.

JEE Main 12

Potential at midpoint of two equal masses separated a?

Show Answer
-4GM/a.

JEE Main 13

Field can be zero but potential nonzero?

Show Answer
Yes, midpoint of equal masses.

JEE Main 14

V due to shell inside?

Show Answer
Constant -GM/R.

JEE Main 15

Work along equipotential?

Show Answer
Zero.

JEE Main 16

Potential change from R to 2R?

Show Answer
Delta V = -GM/(2R)-(-GM/R)=GM/(2R).

JEE Main 17

Energy to move m from R to 2R slowly?

Show Answer
m delta V=GMm/(2R).

JEE Main 18

Total circular orbit energy?

Show Answer
-GMm/(2r).

JEE Main 19

Orbital potential energy?

Show Answer
-GMm/r.

JEE Main 20

Orbital kinetic energy?

Show Answer
GMm/(2r).

JEE Main 21

Escape speed compared to circular speed at same r?

Show Answer
ve=sqrt2 vc.

JEE Main 22

If M doubles, V changes?

Show Answer
Doubles in magnitude, remains negative.

JEE Main 23

If r doubles, V changes?

Show Answer
Half in magnitude.

JEE Main 24

If test mass doubles, V changes?

Show Answer
No, potential is per unit mass.

JEE Main 25

If test mass doubles, U changes?

Show Answer
Doubles.

JEE Main 26

At infinity U?

Show Answer
Zero.

JEE Main 27

At infinity V?

Show Answer
Zero.

JEE Main 28

Potential gradient relates to?

Show Answer
Gravitational field.

JEE Main 29

Field direction relative to potential?

Show Answer
Towards decreasing potential for gravitational field convention.

JEE Main 30

Potential is continuous at shell surface?

Show Answer
Yes.

JEE Main 31

Field inside shell?

Show Answer
Zero.

JEE Main 32

Potential inside shell?

Show Answer
Constant.

JEE Main 33

Why no work inside shell?

Show Answer
Potential is constant, field is zero.

JEE Main 34

Potential at centre of ring qualitatively?

Show Answer
Scalar sum from all mass elements; negative finite.

JEE Main 35

Potential due to symmetric masses can cancel?

Show Answer
Not for positive masses with infinity reference; all terms negative.

JEE Main 36

Potential at square centre if one mass removed?

Show Answer
-3sqrt2 GM/a.

JEE Main 37

Potential energy of system of two masses M and 2M at r?

Show Answer
-2GM2/r.

JEE Main 38

Potential energy of three pair system?

Show Answer
Sum potential energy over all pairs.

JEE Main 39

For three equal masses at triangle side a, system U?

Show Answer
Three pairs, U=-3Gm2/a.

JEE Main 40

For four equal square masses side a, number of pairs?

Show Answer
Six pairs.

JEE Main 41

Potential well represents?

Show Answer
Negative binding potential.

JEE Main 42

To escape, total energy must be?

Show Answer
At least zero.

JEE Main 43

Escape kinetic energy at surface?

Show Answer
GMm/R.

JEE Main 44

If radius doubles and M same, escape energy?

Show Answer
Halves.

JEE Main 45

If M doubles and R same, escape energy?

Show Answer
Doubles.

JEE Main 46

Unit of escape energy?

Show Answer
Joule.

JEE Main 47

Unit of potential?

Show Answer
J kg-1.

JEE Main 48

Potential due to Earth at centre using point formula?

Show Answer
Invalid inside extended sphere; use sphere formula.

JEE Main 49

Potential at surface from inside formula?

Show Answer
Put r=R to get -GM/R.

JEE Main 50

Potential at centre from inside formula compared to surface?

Show Answer
1.5 times surface magnitude, more negative.

JEE Advanced Question Bank

These focus on potential graphs, escape energy, superposition, sphere potential and combined concepts.

JEE Advanced 1

Derive V outside a point mass from field.

Show Answer
V=- integral infinity to r of field dot dr = -GM/r with zero at infinity.

JEE Advanced 2

Show U=mV for test mass.

Show Answer
Potential is potential energy per unit mass, so U=mV.

JEE Advanced 3

Why potential can be nonzero where field is zero?

Show Answer
Field is gradient of potential; potential value can be nonzero even if local gradient is zero.

JEE Advanced 4

Midpoint of two equal masses: field and potential?

Show Answer
Field zero, potential -4GM/a if separation is a.

JEE Advanced 5

Find potential at square centre due to masses m at all corners.

Show Answer
Distance a/sqrt2, V=-4sqrt2 Gm/a.

JEE Advanced 6

Find system energy of three equal masses in equilateral triangle.

Show Answer
U=-3Gm2/a.

JEE Advanced 7

Find system energy of four equal masses at square corners.

Show Answer
Four side pairs: -4Gm2/a. Two diagonal pairs: -2Gm2/(a sqrt2). Total = -(4+sqrt2)Gm2/a.

JEE Advanced 8

Potential at centre of uniform sphere?

Show Answer
-3GM/(2R).

JEE Advanced 9

Potential at r=R/2 inside sphere?

Show Answer
-11GM/(8R).

JEE Advanced 10

Difference between centre and surface potential?

Show Answer
Vc-Vs=-GM/(2R).

JEE Advanced 11

Energy to move mass m from centre to surface of uniform sphere?

Show Answer
m(Vs-Vc)=GMm/(2R).

JEE Advanced 12

Graph continuity at r=R?

Show Answer
Inside and outside potentials both give -GM/R at r=R.

JEE Advanced 13

Slope of potential graph relates to?

Show Answer
Negative of gravitational field radial component.

JEE Advanced 14

Where is potential most negative inside uniform sphere?

Show Answer
At centre.

JEE Advanced 15

Escape speed from height h?

Show Answer
sqrt(2GM/(R+h)).

JEE Advanced 16

Escape energy from radius r?

Show Answer
GMm/r.

JEE Advanced 17

Energy needed from circular orbit to escape?

Show Answer
Total orbit energy is -GMm/(2r), so add GMm/(2r).

JEE Advanced 18

Speed boost from circular orbit to escape speed?

Show Answer
ve-vc=(sqrt2-1)sqrt(GM/r).

JEE Advanced 19

If total energy is positive, path is?

Show Answer
Unbound escape trajectory.

JEE Advanced 20

If total energy is zero, path is?

Show Answer
Parabolic escape limit.

JEE Advanced 21

If total energy is negative, path is?

Show Answer
Bound orbit for two-body gravitational system.

JEE Advanced 22

Potential at centre of regular polygon equal masses?

Show Answer
-nGM/R where R is distance from centre to each mass.

JEE Advanced 23

Field at centre of regular polygon?

Show Answer
Zero by symmetry, while potential is nonzero.

JEE Advanced 24

Potential due to two masses M and 4M, zero field point potential?

Show Answer
At x=d/3 from M. V=-GM/(d/3)-4GM/(2d/3)=-3GM/d-6GM/d=-9GM/d.

JEE Advanced 25

Can potential be chosen with different zero?

Show Answer
Yes, only potential differences matter, but school gravitation commonly sets zero at infinity.

JEE Advanced 26

How does changing zero affect field?

Show Answer
It does not affect field because field depends on gradient.

JEE Advanced 27

Potential inside shell?

Show Answer
Constant -GM/R.

JEE Advanced 28

Work to move inside shell?

Show Answer
Zero because potential difference is zero.

JEE Advanced 29

Potential energy of system of masses formula?

Show Answer
Sum over distinct pairs: U=-sum Gmimj/rij.

JEE Advanced 30

Why not double count pairs?

Show Answer
Each interaction pair contributes once to system potential energy.

JEE Advanced 31

Potential at infinity due to finite masses?

Show Answer
Zero with standard reference.

JEE Advanced 32

Potential energy at infinity for two masses?

Show Answer
Zero.

JEE Advanced 33

Potential at r tends to negative infinity as r tends to?

Show Answer
Zero for ideal point mass.

JEE Advanced 34

Why point-mass potential singularity is idealized?

Show Answer
Real bodies have finite size and mass distribution.

JEE Advanced 35

Potential of uniform sphere finite because?

Show Answer
Mass is distributed over finite volume.

JEE Advanced 36

Energy to move from R to infinity?

Show Answer
GMm/R.

JEE Advanced 37

Energy to move from 2R to infinity?

Show Answer
GMm/(2R).

JEE Advanced 38

Energy to move from R to 2R?

Show Answer
GMm/(2R).

JEE Advanced 39

Potential difference V(2R)-V(R)?

Show Answer
GM/(2R).

JEE Advanced 40

Sign of work by gravity moving outward?

Show Answer
Negative.

JEE Advanced 41

Sign of external work moving outward slowly?

Show Answer
Positive.

JEE Advanced 42

Potential graph for point mass concavity?

Show Answer
It rises toward zero with decreasing slope magnitude.

JEE Advanced 43

Can equipotential surfaces intersect?

Show Answer
No, one point cannot have two potentials.

JEE Advanced 44

Are equipotential surfaces closer where field is stronger?

Show Answer
Yes, potential changes faster where field magnitude is large.

JEE Advanced 45

Field on equipotential surface direction?

Show Answer
Normal to the surface.

JEE Advanced 46

Potential energy of satellite in circular orbit if kinetic energy K?

Show Answer
U=-2K.

JEE Advanced 47

Total energy in circular orbit in terms of K?

Show Answer
E=-K.

JEE Advanced 48

Escape energy from circular orbit in terms of K?

Show Answer
K, because total energy is -K.

JEE Advanced 49

Potential due to continuous body requires?

Show Answer
Integration over mass elements.

JEE Advanced 50

Main advanced trap?

Show Answer
Confusing potential superposition with vector field cancellation.

IB / IGCSE / A-Level Questions

25 IB Questions

  1. Define gravitational potential.
    Show Answer
    Work done per unit mass from infinity.
  2. Unit of V?
    Show Answer
    J kg-1.
  3. Define U.
    Show Answer
    Gravitational potential energy.
  4. Formula for U?
    Show Answer
    -GMm/r.
  5. Formula for V?
    Show Answer
    -GM/r.
  6. Why negative?
    Show Answer
    Zero at infinity and attraction.
  7. Scalar or vector?
    Show Answer
    Scalar.
  8. What is equipotential?
    Show Answer
    Surface of constant potential.
  9. Work on equipotential?
    Show Answer
    Zero.
  10. Escape velocity?
    Show Answer
    sqrt(2GM/R).
  11. Escape energy?
    Show Answer
    GMm/R.
  12. Does escape speed depend on projectile mass?
    Show Answer
    No.
  13. Potential due to many masses?
    Show Answer
    Scalar sum.
  14. Sphere outside?
    Show Answer
    -GM/r.
  15. Sphere surface?
    Show Answer
    -GM/R.
  16. Sphere centre?
    Show Answer
    -3GM/(2R).
  17. Bound system energy sign?
    Show Answer
    Negative.
  18. Potential at infinity?
    Show Answer
    Zero.
  19. Potential well means?
    Show Answer
    Negative binding region.
  20. U=mV?
    Show Answer
    Yes for test mass m.
  21. Field relation?
    Show Answer
    Field is related to potential gradient.
  22. V vs r shape?
    Show Answer
    Negative curve approaching zero.
  23. Common sign error?
    Show Answer
    Dropping negative sign.
  24. Potential inside shell?
    Show Answer
    Constant.
  25. Field inside shell?
    Show Answer
    Zero.

25 IGCSE Questions

  1. What is gravitational energy?
    Show Answer
    Energy due to position in gravity.
  2. What is work done?
    Show Answer
    Energy transferred by force.
  3. What is potential?
    Show Answer
    Energy per unit mass.
  4. Unit of energy?
    Show Answer
    Joule.
  5. Unit of potential?
    Show Answer
    J/kg.
  6. What is escape?
    Show Answer
    Leaving gravitational field without returning.
  7. Does bigger planet need more escape energy?
    Show Answer
    Usually yes for larger M/R.
  8. Potential is positive or negative near Earth?
    Show Answer
    Negative with zero at infinity.
  9. What is infinity reference?
    Show Answer
    Choosing V=0 far away.
  10. What are equipotentials?
    Show Answer
    Same-potential surfaces.
  11. Work along same potential?
    Show Answer
    Zero.
  12. Potential due to Earth decreases or increases outward?
    Show Answer
    Increases toward zero.
  13. Energy of bound orbit?
    Show Answer
    Negative.
  14. What is a potential well?
    Show Answer
    A negative-energy region.
  15. Potential depends on test mass?
    Show Answer
    No.
  16. Potential energy depends on test mass?
    Show Answer
    Yes.
  17. What is scalar?
    Show Answer
    Quantity with magnitude only.
  18. Is potential scalar?
    Show Answer
    Yes.
  19. Can two potentials add?
    Show Answer
    Yes, algebraically.
  20. Potential at far distance?
    Show Answer
    Near zero.
  21. Escape velocity mass dependence?
    Show Answer
    Independent of projectile mass.
  22. What is satellite energy?
    Show Answer
    Energy of satellite in orbit.
  23. Use red objects why?
    Show Answer
    For visibility in diagrams.
  24. Main mistake?
    Show Answer
    Confusing V and U.
  25. Potential surface shape near point mass?
    Show Answer
    Spherical.

25 A-Level Questions

  1. Derive V=-GM/r.
    Show Answer
    Integrate gravitational field from infinity to r.
  2. Derive U=-GMm/r.
    Show Answer
    Multiply V by m.
  3. Derive escape speed.
    Show Answer
    Set (1/2)mv2=GMm/R.
  4. Why mass cancels?
    Show Answer
    Both kinetic and potential energy contain m.
  5. System energy formula?
    Show Answer
    Sum over distinct pairs.
  6. Square four-mass energy?
    Show Answer
    -(4+sqrt2)Gm2/a.
  7. Potential at field-zero point?
    Show Answer
    Can be nonzero.
  8. Inside sphere potential?
    Show Answer
    -(GM/2R3)(3R2-r2).
  9. Potential continuity?
    Show Answer
    Continuous at r=R.
  10. Field continuity for solid sphere?
    Show Answer
    Continuous at surface.
  11. Potential inside shell?
    Show Answer
    Constant.
  12. Equipotential-field relation?
    Show Answer
    Field is perpendicular.
  13. Potential gradient sign?
    Show Answer
    Field equals negative gradient.
  14. Energy in circular orbit?
    Show Answer
    -GMm/(2r).
  15. Escape energy from orbit?
    Show Answer
    GMm/(2r).
  16. Potential at centre of polygon?
    Show Answer
    -nGM/R.
  17. Zero reference change effect?
    Show Answer
    No effect on field.
  18. Potential well depth?
    Show Answer
    Energy needed to escape per unit mass.
  19. Bound condition?
    Show Answer
    Total energy negative.
  20. Unbound condition?
    Show Answer
    Total energy positive.
  21. Parabolic condition?
    Show Answer
    Total energy zero.
  22. U and V relation?
    Show Answer
    U=mV.
  23. Potential due to continuous mass?
    Show Answer
    Integrate -G dm/r.
  24. Why potential easier than field sometimes?
    Show Answer
    Scalar addition.
  25. Main advanced trap?
    Show Answer
    Using vector cancellation for potential.

Assertion Reason Questions

AR 1

Assertion: Gravitational potential is scalar. Reason: It is work per unit mass.

Show Answer
Both true and reason supports assertion.

AR 2

Assertion: V is negative near Earth. Reason: Zero is taken at infinity.

Show Answer
Both true and reason explains assertion.

AR 3

Assertion: Work along equipotential is zero. Reason: Potential difference is zero.

Show Answer
Both true and reason explains assertion.

AR 4

Assertion: Escape velocity depends on projectile mass. Reason: Escape energy contains m.

Show Answer
Assertion false; reason true but m cancels in velocity.

AR 5

Assertion: Potential can be nonzero where field is zero. Reason: Field depends on potential gradient.

Show Answer
Both true and reason explains assertion.

AR 6

Assertion: Potential inside shell is constant. Reason: Field inside shell is zero.

Show Answer
Both true and reason explains assertion.

AR 7

Assertion: Potential energy is always positive. Reason: Energy is scalar.

Show Answer
Assertion false; reason true.

AR 8

Assertion: Potential due to system is vector sum. Reason: Potential has direction.

Show Answer
Both false.

AR 9

Assertion: Field lines are perpendicular to equipotentials. Reason: Tangential work on equipotential is zero.

Show Answer
Both true and reason explains assertion.

AR 10

Assertion: V outside sphere is -GM/r. Reason: Sphere behaves as point mass externally.

Show Answer
Both true and reason explains assertion.

AR 11

Assertion: V at sphere centre is finite. Reason: Sphere has finite size.

Show Answer
Both true.

AR 12

Assertion: Point mass potential becomes singular at r=0. Reason: Formula has r in denominator.

Show Answer
Both true.

AR 13

Assertion: Total circular orbit energy is negative. Reason: Orbit is bound.

Show Answer
Both true.

AR 14

Assertion: Escape total energy is zero at threshold. Reason: At infinity both K and U can be zero.

Show Answer
Both true and reason explains assertion.

AR 15

Assertion: U=mV. Reason: Potential is energy per unit mass.

Show Answer
Both true and reason explains assertion.

AR 16

Assertion: V and U are identical. Reason: Both involve gravity.

Show Answer
Both false as stated.

AR 17

Assertion: Potential at infinity is zero by convention. Reason: Only potential difference matters.

Show Answer
Both true.

AR 18

Assertion: Changing zero of potential changes field. Reason: Field is gradient.

Show Answer
Assertion false; reason true.

AR 19

Assertion: Escape energy depends on m. Reason: More massive projectile needs more energy.

Show Answer
Both true and reason explains assertion.

AR 20

Assertion: Escape speed does not depend on m. Reason: m cancels from energy equation.

Show Answer
Both true and reason explains assertion.

AR 21

Assertion: Potential graph approaches zero at large r. Reason: V=-GM/r.

Show Answer
Both true.

AR 22

Assertion: Potential inside uniform sphere is parabolic in r. Reason: Formula contains r2.

Show Answer
Both true.

AR 23

Assertion: Potential is discontinuous at surface. Reason: Formula changes there.

Show Answer
Assertion false; reason true but formulas match.

AR 24

Assertion: Equipotential surfaces can intersect. Reason: Potential is scalar.

Show Answer
Assertion false; reason true.

AR 25

Assertion: Potential due to masses adds algebraically. Reason: Potential is scalar.

Show Answer
Both true and reason explains assertion.

AR 26

Assertion: Potential well means positive energy hill. Reason: Gravity is attractive.

Show Answer
Assertion false; reason true.

AR 27

Assertion: Bound systems have negative total energy. Reason: They cannot escape without added energy.

Show Answer
Both true.

AR 28

Assertion: Potential energy of a pair is counted once. Reason: Interaction pair is mutual.

Show Answer
Both true.

AR 29

Assertion: Potential at midpoint of equal masses is zero. Reason: Fields cancel there.

Show Answer
Assertion false; reason true.

AR 30

Assertion: Red objects are used in diagrams. Reason: They improve visibility on light backgrounds.

Show Answer
Both true for this notes page design.

Case Study Questions

Potential Wells

Passage: A planet creates a negative gravitational potential well around it.

  1. What is the zero reference?
  2. What does deeper well mean?
  3. What energy is needed to escape?
Show Answer

Answers: Infinity; stronger binding; energy equal to well depth. Explanation: Negative potential measures bound state relative to infinity.

Satellites

Passage: A satellite in circular orbit has kinetic and potential energy.

  1. What is U?
  2. What is total E?
  3. Is satellite bound?
Show Answer

Answers: -GMm/r; -GMm/(2r); yes, E is negative.

Escape Velocity

Passage: A projectile is launched from Earth's surface with minimum escape speed.

  1. Formula for ve?
  2. Formula for escape energy?
  3. Does speed depend on projectile mass?
Show Answer

Answers: sqrt(2GM/R); GMm/R; no. Explanation: Mass cancels in velocity equation.

Earth-Moon System

Passage: Earth and Moon have mutual gravitational potential energy.

  1. Write U.
  2. Why negative?
  3. What happens as distance increases?
Show Answer

Answers: -GMm/r; attraction with zero at infinity; U increases toward zero.

Planetary Systems

Passage: Planets remain bound to stars because total mechanical energy is negative.

  1. Sign of U?
  2. Sign of total E for bound orbit?
  3. What if total E becomes positive?
Show Answer

Answers: Negative; negative; planet escapes in an unbound path.

Common Student Mistakes

Confusing V and U

V is potential per unit mass. U is potential energy and includes the test mass.

Sign Mistakes

With infinity as zero, gravitational potential and potential energy are negative near a mass.

Infinity Reference Errors

Do not set zero at Earth's surface unless the question clearly changes the reference.

Sphere Formula Confusion

Outside, surface and inside sphere formulas are different but must match at r=R.

Equipotential Misconceptions

No work is done along an equipotential surface, and field lines are perpendicular to it.

Scalar Superposition Error

Potential does not cancel like vectors. Add scalar values with signs.

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