GRE Physics Companion: Fixed-Axis Rotation and Particle Kinematics

GRE Physics Companion: Fixed-Axis Rotation and Particle Kinematics

For a material point at perpendicular radius ρ,

(1)

and

(2)

The corresponding magnitudes are

(3)

1 High-value GRE facts

1. Use perpendicular distance to the axis, not distance to an arbitrary origin.
2. All points share the same ω and α, but not the same v or a.
3. velocity is tangent to the circular path.
4. normal acceleration points toward the axis.
5. tangential acceleration changes the speed.
6. a_n contains ω².
7. A point displaced along the axis has the same kinematics as a point with the same perpendicular radius.
8. The rotating basis vectors must be differentiated.
9. The double-cross-product acceleration is inward.
10. Relative velocity between two points is perpendicular to their separation vector.

Part I: Original GRE-style problems

Problem 1: speed at a known radius

A point lies 0.30 m from a fixed axis and the body rotates at 10 rad∕s. The point’s speed is

1. 0.3 m∕s
2. 1.0 m∕s
3. 3.0 m∕s
4. 10 m∕s
5. 30 m∕s

Problem 2: normal acceleration

For the point in Problem 1, the normal acceleration magnitude is

1. 3 m∕s²
2. 10 m∕s²
3. 30 m∕s²
4. 100 m∕s²
5. 300 m∕s²

Problem 3: tangential acceleration

If the same body has α = 4 rad∕s², the tangential acceleration magnitude at ρ = 0.30 m is

1. 0.3 m∕s²
2. 0.75 m∕s²
3. 1.2 m∕s²
4. 4.0 m∕s²
5. 12 m∕s²

Problem 4: two radii

Two points on the same rigid body lie at radii r and 4r. Their speed ratio is

1. 1∕4
2. 1
3. 2
4. 4
5. 16

Problem 5: same two radii

For the points in Problem 4, their normal-acceleration ratio at the same instant is

1. 1∕4
2. 1
3. 2
4. 4
5. 16

Problem 6: axial offset

Two material points have equal perpendicular radius ρ but different positions along the fixed axis. Their speed magnitudes are

1. always equal
2. proportional to their axial separation
3. inversely proportional to their axial separation
4. equal only if α = 0
5. zero

Problem 7: direction of velocity

The instantaneous velocity of a point in fixed-axis rotation is

1. radial outward
2. radial inward
3. tangent to the circular path
4. parallel to the axis
5. always zero

Problem 8: constant angular speed

If ω is constant and nonzero, the particle acceleration is

1. zero
2. purely tangential
3. purely inward normal
4. parallel to velocity
5. outward

Problem 9: vector velocity

If ω = 5e_z rad∕s and r = 2e_x m, then

1. v = 10e_x m∕s
2. v = −10e_x m∕s
3. v = 10e_y m∕s
4. v = −10e_y m∕s
5. v = 0

Problem 10: relative velocity rigidity check

For two points A and B on the same rigid body,

(4)

is

1. positive
2. negative
3. zero
4. equal to ω
5. equal to α

Problem 11: inverse relation

A point at radius 0.25 m has tangential speed 5 m∕s. The angular speed magnitude is

1. 1.25 rad∕s
2. 5 rad∕s
3. 10 rad∕s
4. 20 rad∕s
5. 25 rad∕s

Problem 12: kinetic-energy bridge

For one particle in fixed-axis rotation, its kinetic energy can be written

1. mρω
2. mρω²
3. mρ²ω²
4. mρ²ω
5. mω²∕ρ

Part II: Complete worked solutions

Solution 1

(5)

Answer: (C).

Solution 2

(6)

Answer: (C).

Solution 3

(7)

Answer: (C).

Solution 4

Since v = ρ|ω| and both points share the same ω,

(8)

Answer: (D).

Solution 5

Since a_n = ρω² at the same instant,

(9)

Answer: (D).

Solution 6

Fixed-axis kinematics depends on perpendicular radius, not axial position. Equal ρ and common ω give equal speeds. Answer: (A).

Solution 7

The velocity is in the e_𝜃 direction, tangent to the circular path. Answer: (C).

Solution 8

Constant angular speed implies α = 0, so only

(10)

remains. Answer: (C).

Solution 9

v	= ω × r	(11)
	= 5ez × 2ex	(12)
	= 10ey m∕s.	(13)

Answer: (C).

Solution 10

Rigid body relative velocity is perpendicular to separation:

(14)

Answer: (C).

Solution 11

(15)

Answer: (D).

Solution 12

Since v = ρ|ω|,

(16)

Answer: (C).

2 GRE checklist

1. Identify the perpendicular radius first.
2. Keep e_r and e_𝜃 directions straight.
3. Velocity is tangential, not radial.
4. Use a_n = ρω² and a _t = ρ|α|.
5. Equal angular speed does not imply equal linear speed at different radii.
6. Axial offset does not change fixed-axis kinematics at the same ρ.
7. Use cross products for vector direction questions.
8. Check rigidity with r_B∕A ⋅ v_B∕A = 0.

References

References