GRE Physics Companion: Energy Methods for Particle Systems

GRE Physics Companion: Energy Methods for Particle Systems

The two formulas to recognize immediately are

(1)

and

(2)

For two particles,

(3)

1 High-value GRE facts

1. Total kinetic energy is the sum of the kinetic energies of all particles.
2. K = K_CM + K_rel.
3. internal forces cancel from the net force but not generally from work.
4. Actual external work uses the displacement of each force application point.
5. F_ext,net ⋅ dR_CM is pseudowork and tracks K_CM.
6. Zero total momentum does not imply zero total kinetic energy.
7. If no net external force acts, V _CM is constant.
8. An explosion can increase K_rel while leaving K_CM unchanged.
9. For two particles, use reduced mass for relative kinetic energy.
10. In uniform gravity, the system potential is MgY _CM.

Part I: Original GRE-style problems

Problem 1: center-of-mass velocity

Two particles move along a line. Particle 1 has mass 2m and velocity +v, while particle 2 has mass m and velocity −v. The center-of-mass velocity is

1. −v
2. −v∕3
3. 0
4. +v∕3
5. +v

Problem 2: zero momentum

A system has total momentum zero. Which statement must be true?

1. Its total kinetic energy is zero.
2. Its center-of-mass kinetic energy is zero.
3. Every particle is at rest.
4. Its relative kinetic energy is zero.
5. No internal forces act.

Problem 3: kinetic-energy decomposition

A system has total kinetic energy 50 J and center-of-mass kinetic energy 18 J. Its relative kinetic energy is

1. 18 J
2. 32 J
3. 50 J
4. 68 J
5. 900 J

Problem 4: reduced mass

Two particles have masses m and 3m. Their reduced mass is

1. m∕4
2. 3m∕4
3. m
4. 3m
5. 4m

Problem 5: internal force work

Which statement is correct for equal-and-opposite internal forces?

1. Their net internal work is always zero.
2. They cancel from the net force, but their total work need not vanish.
3. They can never change relative kinetic energy.
4. They are always nonconservative.
5. They cannot store potential energy.

Problem 6: explosion at rest

A system initially at rest explodes into two fragments in the absence of external impulse. Which quantity must remain zero?

1. total kinetic energy
2. relative kinetic energy
3. center-of-mass velocity
4. speed of each fragment
5. internal energy change

Problem 7: actual work versus pseudowork

A constant external force F moves one particle of a two-equal-particle system through distance d, while the second particle remains fixed. The actual external work is Fd. The center of mass moves

1. 0
2. d∕4
3. d∕2
4. d
5. 2d

Problem 8: pseudowork

For the system in Problem 7, the center-of-mass pseudowork is

1. 0
2. Fd∕4
3. Fd∕2
4. Fd
5. 2Fd

Problem 9: uniform gravity

For particles in a uniform gravitational field,

(4)

This equals

1. Mg∕Y _CM
2. MgY _CM
3. M²gY _CM
4. gY _CM∕M
5. zero

Problem 10: two-fragment momentum

A 2 kg fragment moves at +6 m∕s after an explosion from rest. The second fragment has mass 3 kg. Its velocity is

1. −9 m∕s
2. −6 m∕s
3. −4 m∕s
4. +4 m∕s
5. +9 m∕s

Problem 11: explosion kinetic energy

For the fragments in Problem 10, the total kinetic energy is

1. 12 J
2. 24 J
3. 36 J
4. 60 J
5. 72 J

Problem 12: isolated spring pair

Two particles connected by an ideal spring form an isolated system. Which quantity is necessarily constant?

1. each particle’s kinetic energy
2. the spring potential energy
3. K_rel
4. K_CM
5. each particle’s speed

Part II: Complete worked solutions

Solution 1

The total mass is

(5)

The total momentum is

(6)

Thus

(7)

Answer: (D).

Solution 2

If total momentum is zero,

(8)

Thus

(9)

and

(10)

The particles can still move relative to one another.

Answer: (B).

Solution 3

Use

(11)

Thus

(12)

Answer: (B).

Solution 4

The reduced mass is

μ	=	(13)
	=	(14)
	= m.	(15)

Answer: (B).

Solution 5

Newton’s third law makes the pair forces equal and opposite, so they cancel from the net system force.

Their work is

(16)

which need not vanish.

Answer: (B).

Solution 6

With no external impulse, total momentum is constant.

The initial system is at rest, so

(17)

and therefore

(18)

after the explosion.

Answer: (C).

Solution 7

For equal masses,

(19)

If particle 1 moves by d and particle 2 remains fixed,

(20)

Answer: (C).

Solution 8

Pseudowork is

(21)

Using

(22)

we obtain

(23)

Answer: (C).

Solution 9

By definition,

(24)

Therefore

(25)

Multiplying by g,

(26)

Answer: (B).

Solution 10

Momentum conservation gives

(27)

Thus

(28)

Answer: (C).

Solution 11

The kinetic energy is

K	= (2)(62) + (3)(42)	(29)
	= 36 + 24	(30)
	= 60 J.	(31)

Answer: (D).

Solution 12

An isolated system has no net external force, so

(32)

is constant.

Therefore

(33)

is constant.

The spring can exchange energy with relative motion, so K_rel and U_s need not be individually constant.

Answer: (D).

2 GRE checklist

For particle-system energy problems:

1. Compute the center-of-mass velocity when momentum information is available.
2. Use K = K_CM + K_rel.
3. For two particles, consider the reduced mass.
4. Do not assume internal work vanishes.
5. Distinguish actual external work from pseudowork.
6. Use momentum conservation for explosions and internal releases.
7. In uniform gravity, replace ∑ m_igy_i with MgY _CM when useful.
8. State the system boundary before assigning internal and external interactions.

References

References