GRE Physics Companion: Coordinate Systems for Mechanics

This companion entry is attached to M00-05: Coordinate Systems for Mechanics. The core entry develops coordinate systems as part of normal mechanics instruction. This companion isolates the rapid recognition and conversion skills useful in timed GRE-style questions.

1 Fast coordinate-choice strategy

Before converting any coordinates, ask what geometry the problem is already giving you.

Useful recognition rules are:

• straight-line, projectile, and rectangular geometry usually favor Cartesian coordinates;
• planar circles and rotation about an axis usually favor polar or cylindrical coordinates;
• inverse-square and other point-centered central-force problems usually favor spherical coordinates;
• an incline often favors a rotated Cartesian basis rather than a curvilinear system.

2 Fast formulas worth recognizing

For Plane polar coordinates,

(1)

and

(2)

For cylindrical coordinates,

(3)

with z unchanged.

For the spherical convention used in M00-05,

(4)

so a point in the equatorial plane has

(5)

For a small arc at fixed radius,

(6)

with the angle in radians.

3 Worked GRE example 1: arc length without coordinate conversion

A particle moves on a circle of radius 2.00 m through an angle of 30^∘. Find the path length.

Convert the angular change to radians:

(7)

At fixed radius,

(8)

so

(9)

Thus

(10)

No Cartesian conversion was needed.

4 Worked GRE example 2: recognize a central-force coordinate

A force is known to point toward the origin and to have magnitude proportional to 1∕r². Which coordinate system most directly exposes the force geometry?

The force depends only on distance from one point and points along the radial direction. Spherical coordinates therefore make the force take the one-component form

(11)

The correct coordinate choice is spherical.

5 GRE-speed questions

MECH-GRE-COORD-001

The point (−3, 3) in the xy plane has polar angle

1. 45^∘
2. 90^∘
3. 135^∘
4. 225^∘

MECH-GRE-COORD-002

The differential volume element in cylindrical coordinates is

1. dρdϕdz
2. ρdρdϕdz
3. ρ² dρdϕdz
4. r² sin 𝜃 dr d𝜃 dϕ

MECH-GRE-COORD-003

Using the convention that 𝜃 is measured from +z, a point in the xy plane with r > 0 has

1. 𝜃 = 0^∘
2. 𝜃 = 45^∘
3. 𝜃 = 90^∘
4. 𝜃 = 180^∘

MECH-GRE-COORD-004

A particle moves around a circle at fixed r with increasing 𝜃. Its instantaneous direction of motion is along

1. e_r
2. −e_r
3. e_𝜃
4. −e_𝜃

6 Answers and brief rationales

1. C. The point is in quadrant II, so the polar angle is 135^∘.
2. B. The azimuthal physical distance is ρdϕ, producing the Jacobian factor ρ.
3. C. The equatorial plane is perpendicular to +z, so the polar angle is 90^∘.
4. C. At fixed radius, increasing 𝜃 points along the transverse unit vector e_𝜃.

7 Timed-test cautions

1. Do not convert coordinates unless the requested quantity requires it.
2. Use quadrant information when reconstructing angles from Cartesian components.
3. Check the spherical-angle convention before using a memorized formula.
4. Use radians in differential arc formulas.
5. Remember that changing coordinate systems does not by itself change the physical reference frame.

References