GRE Physics Companion: Angular Velocity and Angular Acceleration as Vectors

GRE Physics Companion: Angular Velocity and Angular Acceleration as Vectors

The central rigid body kinematics relations are

(1)

and

(2)

For rotation about a fixed point,

(3)

1 High-value GRE facts

1. ω points along the instantaneous rotation axis by the right-hand rule.
2. Finite three-dimensional rotations do not generally commute.
3. Infinitesimal rotations behave vectorially to first order.
4. v = ω ×ρ.
5. α = dω∕dt can be nonzero even when |ω| is constant.
6. The normal acceleration term is ω × (ω ×ρ).
7. For a rigid body, one common ω relates every pair of points.
8. v_B − v_A is perpendicular to r_B∕A.
9. [ω×] is skew-symmetric.
10. Only the component of α parallel to ω changes angular speed.

Part I: Original GRE-style problems

Problem 1: right-hand rule

A disk rotates counterclockwise when viewed from the +z side. Its angular velocity vector points

1. +x
2. −x
3. +y
4. +z
5. −z

Problem 2: point velocity

Let

(4)

and

(5)

Then the point velocity is

1. 6e_x m∕s
2. −6e_x m∕s
3. 6e_y m∕s
4. −6e_y m∕s
5. zero

Problem 3: point on the axis

If ρ is parallel to ω, then the rotational velocity ω ×ρ is

1. parallel to ω
2. perpendicular to ω with magnitude ωρ
3. zero
4. equal to ρ
5. undefined

Problem 4: changing direction

A body has constant angular speed but the direction of ω changes. Which statement is correct?

1. α = 0
2. α must be parallel to ω
3. α can be nonzero
4. the body cannot be rigid
5. the angular speed must increase

Problem 5: normal acceleration

For fixed angular velocity and position ρ perpendicular to the axis,

(6)

points

1. along ρ
2. opposite ρ
3. along ω
4. tangent to the circle
5. out of the plane at random

Problem 6: relative velocity

Points A and B on a rigid body satisfy

(7)

Then

(8)

is

1. 8e_x m∕s
2. −8e_x m∕s
3. 8e_y m∕s
4. −8e_y m∕s
5. zero

Problem 7: rigidity condition

For two points on a rigid body, the relative velocity v_B − v_A is always

1. parallel to r_B∕A
2. perpendicular to r_B∕A
3. zero
4. parallel to ω
5. equal to r_B∕A

Problem 8: skew matrix

The matrix representing cross product with ω satisfies

1. [ω×]^T = [ω×]
2. [ω×]^T = −[ω×]
3. [ω×]² = 0 always
4. its determinant is always one
5. it is diagonal

Problem 9: angular-speed derivative

If α ⊥ ω at an instant, then at that instant

1. angular speed is increasing
2. angular speed is decreasing
3. ω = 0
4. ω = 0
5. α = 0

Problem 10: finite rotations

Two finite rotations about different axes are generally

1. commutative
2. noncommutative
3. always equivalent to scalar addition
4. independent of order only when both are 90^∘
5. impossible for a rigid body

Problem 11: acceleration with zero angular acceleration

If α = 0 but ω≠0, a point away from the axis has

1. zero acceleration
2. only the normal double-cross-product acceleration
3. only tangential acceleration
4. acceleration parallel to velocity
5. arbitrary acceleration unrelated to rotation

Problem 12: translating reference point

For a translating and rotating rigid body, the velocity of point B is

1. ω × r_B∕A only
2. v_A + ω × r_B∕A
3. v_A + α× r_B∕A
4. v_A + ω²r _B∕A
5. always equal to v_A

Part II: Complete worked solutions

Solution 1

Counterclockwise rotation viewed from +z corresponds to the right-hand thumb pointing along +z.

Answer: (D).

Solution 2

v	= ω ×ρ	(9)
	= (3ez) × (2ex)	(10)
	= 6ey m∕s.	(11)

Answer: (C).

Solution 3

Parallel vectors have zero cross product:

(12)

Answer: (C).

Solution 4

angular acceleration is the vector derivative

(13)

A changing direction produces a nonzero derivative even if the magnitude remains fixed.

Answer: (C).

Solution 5

For ρ ⊥ ω,

(14)

Answer: (B).

Solution 6

vB − vA	= ω × rB∕A	(15)
	= (4ez) × (2ex)	(16)
	= 8ey m∕s.	(17)

Answer: (C).

Solution 7

The rigid distance constraint gives

(18)

Thus the relative velocity is perpendicular to the separation vector.

Answer: (B).

Solution 8

The cross-product matrix is skew-symmetric:

(19)

Answer: (B).

Solution 9

(20)

If the vectors are perpendicular, their dot product is zero.

Answer: (C).

Solution 10

Finite rotations about different axes generally depend on order.

Answer: (B).

Solution 11

With α = 0,

(21)

Answer: (B).

Solution 12

For two points on the same rigid body,

(22)

Answer: (B).

2 GRE checklist

1. Apply the right-hand rule before doing component algebra.
2. Preserve cross-product order.
3. Distinguish angular speed from angular velocity vector.
4. Remember that direction change of ω contributes to α.
5. For acceleration, include both α× r and the double-cross-product term.
6. For translating bodies, include the reference-point velocity or acceleration.
7. Use the rigid body orthogonality check r_B∕A ⋅ (v_B − v_A) = 0.
8. Recognize [ω×] as a skew-symmetric matrix.

References

References