GRE Physics Companion: Angular Momentum of Particle Systems

GRE Physics Companion: Angular Momentum of Particle Systems

The core relations are

(1)

(2)

and

(3)

For a particle system with central internal forces,

(4)

1 High-value GRE facts

1. Angular momentum depends on the chosen origin.
2. L = r_⊥p for a particle.
3. Torque is the time rate of change of angular momentum.
4. Zero external torque implies constant total angular momentum.
5. angular impulse equals ΔL.
6. A central force has zero torque about its force center.
7. In the center-of-mass decomposition, L_O = R_CM × P + L_CM.
8. Radial motion contributes no angular momentum about the center.
9. For planar polar motion, L = mr²𝜃.
10. For fixed-axis rigid body rotation, L_z = I_zω.

Part I: Original GRE-style problems

Problem 1: perpendicular motion

A particle has momentum magnitude p and moves along a line whose perpendicular distance from the origin is b. Its angular-momentum magnitude about the origin is

1. p∕b
2. bp
3. b∕p
4. p²b
5. zero

Problem 2: radial motion

A particle moves directly away from the origin. Its angular momentum about the origin is

1. mrv
2. mrv∕2
3. mv∕r
4. zero
5. dependent on acceleration

Problem 3: torque

A force F acts parallel to the position vector r. The torque about the origin is

1. rF
2. rF∕2
3. zero
4. F∕r
5. mrv

Problem 4: angular impulse

A constant torque 5 N m acts for 3 s. The change in angular momentum magnitude is

1. 5 kg m²∕s
2. 8 kg m²∕s
3. 15 kg m²∕s
4. 25 kg m²∕s
5. 75 kg m²∕s

Problem 5: central force

A particle moves under a central force. Which quantity is necessarily conserved about the force center?

1. linear momentum
2. angular momentum
3. kinetic energy for every central force
4. speed
5. position

Problem 6: center-of-mass decomposition

A system has R_CM = 2e_x m, total momentum P = 3e_y kg m∕s, and L_CM = 4e_z kg m²∕s. Its angular momentum about the origin is

1. 2e_z
2. 4e_z
3. 6e_z
4. 10e_z
5. 12e_z

Problem 7: planar polar motion

For a particle in planar polar coordinates, angular momentum magnitude about the origin is

1. mṙ
2. mr𝜃
3. mr²𝜃
4. mr²ṙ
5. m𝜃∕r

Problem 8: areal velocity

For a particle of mass m with angular momentum magnitude L under a central force, the areal velocity is

1. L∕m
2. L∕(2m)
3. 2L∕m
4. m∕(2L)
5. zero

Problem 9: origin shift

Two origins differ by constant vector a. If total linear momentum is zero, then the total angular momentum about the two origins is

1. always different
2. identical
3. opposite
4. related by a factor of two
5. undefined

Problem 10: internal forces

For two particles with equal-and-opposite central internal forces, the pair’s total internal torque about any origin is

1. zero
2. twice either individual torque
3. always parallel to the force
4. proportional to total mass
5. nonzero unless the origin is the center of mass

Problem 11: fixed-axis rigid body

A rigid body rotates about a fixed z axis. Its axial angular momentum is

1. Mω
2. I_z∕ω
3. I_zω
4. I_zω²
5. Mr_CMω

Problem 12: zero external torque

If the net external torque on a particle system is zero and internal torques cancel, then

1. every particle velocity is constant
2. total linear momentum must be zero
3. total angular momentum is constant
4. kinetic energy must be constant
5. all forces vanish

Part II: Complete worked solutions

Solution 1

Use the perpendicular-distance form:

(5)

Answer: (B).

Solution 2

Radial motion has r parallel to p, so

(6)

Answer: (D).

Solution 3

If F is parallel to r,

(7)

Answer: (C).

Solution 4

(8)

Answer: (C).

Solution 5

A central force gives zero torque about its center, so angular momentum is conserved. Answer: (B).

Solution 6

LO	= RCM × P + LCM	(9)
	= (2ex) × (3ey) + 4ez	(10)
	= 10ez kg m2∕s.	(11)

Answer: (D).

Solution 7

(12)

Answer: (C).

Solution 8

(13)

Answer: (B).

Solution 9

Use

(14)

If P = 0, the two angular momenta are identical. Answer: (B).

Solution 10

The pair torque is

(15)

For a central force, the two vectors are parallel, so the cross product is zero. Answer: (A).

Solution 11

For fixed-axis rotation,

(16)

Answer: (C).

Solution 12

From

(17)

zero external torque implies constant total angular momentum. Answer: (C).

2 GRE checklist

1. State the origin before computing angular momentum or torque.
2. Use r_⊥p when the geometry is simpler than a full cross product.
3. Apply the right-hand rule to determine vector direction.
4. Connect angular impulse to ΔL.
5. For system problems, distinguish external torque from internal torque.
6. Recognize central forces immediately as zero-torque forces about their center.
7. Use the center-of-mass decomposition when translation and internal motion coexist.
8. Do not use L = Iω as an unrestricted three-dimensional Vector Identity.

References

References