GRE Physics Companion: Cylindrical and Spherical Particle Kinematics

This companion focuses on fast recognition rather than full derivation. For most test problems, first identify the geometry and then use the physically scaled velocity components rather than the raw coordinate rates.

1 Fast coordinate triage

For cylindrical coordinates,

(1)

For the spherical convention used in M01-11,

(2)

The scale factors ρ, r, and r sin 𝜃 are the main quantities students most often omit under time pressure.

2 Common traps

Cylindrical and spherical basis vectors are not generally fixed. Angular motion rotates them even when the coordinate magnitudes are constant.

In spherical coordinates, always confirm the angle convention. Here 𝜃 is measured from +z and ϕ is the azimuth.

The raw rate dϕ∕dt has units of inverse time; the physical azimuthal speed is ρ(dϕ∕dt) in cylindrical coordinates and r sin 𝜃(dϕ∕dt) in spherical coordinates.

3 Worked GRE example 1: cylindrical circular motion

A particle moves with constant cylindrical coordinates

(3)

while

(4)

is constant. Find its speed and acceleration magnitude.

The speed is

(5)

The only nonzero acceleration component is

(6)

Therefore

(7)

This is ordinary uniform circular motion written in cylindrical coordinates.

4 Worked GRE example 2: spherical equatorial motion

A particle has

(8)

with all other coordinate rates zero.

Because sin 90^∘ = 1,

(9)

The radial acceleration is

(10)

and a_𝜃 = a_ϕ = 0. The motion is a circle of radius 3.0 m in the equatorial plane.

5 GRE-speed questions

1. In cylindrical coordinates a particle has ρ = 4 m and dϕ∕dt = 3 rad/s with dρ∕dt = dz∕dt = 0. Its speed is (A) 3 (B) 4 (C) 7 (D) 12 m/s.
2. In spherical coordinates using the convention of M01-11, the physical azimuthal speed is (A) dϕ∕dt (B) r(dϕ∕dt) (C) r sin 𝜃(dϕ∕dt) (D) r cos 𝜃(dϕ∕dt).
3. A particle has fixed ρ and fixed z with constant dϕ∕dt = ω. Its radial acceleration is (A) 0 (B) +ρω² (C) −ρω² (D) 2ρω.
4. Which statement is true? (A) e_ρ is fixed in an inertial Cartesian frame. (B) e_ϕ changes whenever ϕ changes. (C) e_z rotates with ϕ. (D) Spherical e_r is independent of angle.

6 Answers and rationales

1. D. v_ϕ = ρ(dϕ∕dt) = (4)(3) = 12 m/s.
2. C. The radius of the azimuthal circle is r sin 𝜃.
3. C. Constant-radius circular motion has inward acceleration −ρω²e _ρ.
4. B. The cylindrical radial and azimuthal unit vectors rotate with the azimuthal coordinate.

References