Unit 5 · Topic 5.1 Beta

Rotational Kinematics

Rotation about a fixed axis is described by the angle θ in radians, the angular velocity ω = Δθ/Δt and the angular acceleration α = Δω/Δt, each with a sign for clockwise or counterclockwise.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

A washing machine drum spins up, holds a fast spin, then coasts to a stop. Its edge travels a long way, yet the drum never goes anywhere. To describe turning, physics swaps position, velocity and acceleration for angle, angular velocity and angular acceleration, and every equation you already know comes back with new letters.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A cart starts from rest with a = 4.0 m/s². How far does it go in 3.0 s?

  1. 18 m
  2. 12 m
  3. 36 m
  4. 6.0 m
Show the answer

x = v₀t + ½at² = 0 + ½(4.0)(3.0)² = 18 m.

  • Correct: 18 m:
  • 12 m:
  • 36 m:
  • 6.0 m:

2. On a velocity-time graph, the area under the line gives

  1. the displacement
  2. the acceleration
  3. the velocity
  4. the time
Show the answer

Area = velocity × time = displacement; the slope is the acceleration.

  • Correct: the displacement:
  • the acceleration:
  • the velocity:
  • the time:

3. An object goes around a circle once every 2.0 s. Its frequency is

  1. 0.50 Hz
  2. 2.0 Hz
  3. 3.1 Hz
  4. 12.6 Hz
Show the answer

f = 1/T = 1/(2.0 s) = 0.50 revolutions per second.

  • Correct: 0.50 Hz:
  • 2.0 Hz:
  • 3.1 Hz:
  • 12.6 Hz:

Part 4 · See it

See it first

A wheel turns about an axis through its center. A painted spoke has turned counterclockwise from a dashed reference line through an angle of 1 radian: the distance along the rim between the reference line and the spoke, highlighted, equals the radius r. A curved arrow outside the rim shows the angular velocity ω, counterclockwise, called positive. A side panel pairs each straight-line quantity with its rotational partner: position x with angle θ in radians, velocity v with angular velocity ω, acceleration a with angular acceleration α, and the three constant-acceleration equations with their rotational forms. One revolution is 2π rad, or 360 degrees.
A wheel turned through 1 rad: the distance along the rim equals the radius. Each straight-line quantity has a rotational partner. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Every point of a rigid wheel turns together about the axisone angle θ describes the whole wheel
  2. The angle changes over a time intervalthe wheel has an angular velocity ω = Δθ/Δt, in rad/s
  3. The angular velocity changes over a time intervalthe wheel has an angular acceleration α = Δω/Δt, in rad/s²
  4. α is constantthe three kinematic equations hold with θ, ω, α in place of x, v, a
  5. α has the same sign as ω (or the opposite sign)the wheel speeds up (or slows down)

Part 6 · Key ideas

Key ideas

  • Measure angles in radians: θ = (distance along the circle) ÷ r. One revolution = 2π rad = 360°.
  • Pick a positive direction, usually counterclockwise. Then θ, ω and α carry signs exactly as x, v and a do on a line.
  • A rigid system turns as one piece: every point shares the same θ, ω and α.
  • Constant α: ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2αΔθ.
  • On an ω-t graph the slope is α and the area is Δθ.

Part 7 · Misconception

A common mistake

The wrong idea: A point near the rim of a turning wheel has a larger angular velocity than a point near the axle, because it moves faster.

What actually happens: Both points turn through the same angle in the same time, so they share one ω. The point near the rim covers more distance, which is a separate quantity (its speed, next topic).

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

A potter’s wheel speeding up

chalk markTop view: turns counterclockwiseStarts from rest.Constant angular acceleration2.0 rad/s² for 3.0 s.Counterclockwise is positive.

1. What is the wheel’s angular velocity at the end of the 3.0 s?

Type a number in rad/s.

Show the answer

From rest, ω = ω₀ + αt = 0 + (2.0 rad/s²)(3.0 s) = 6.0 rad/s.

  • Answer: 6.0 rad/s

2. Through what angle, in radians, does the chalk mark turn during the 3.0 s?

Type a number in rad.

Show the answer

θ = ω₀t + ½αt² = 0 + ½(2.0 rad/s²)(3.0 s)² = 9.0 rad.

  • Answer: 9.0 rad

3. A second chalk mark is drawn halfway between the first mark and the axle. During the same 3.0 s, its angular displacement is

  1. the same as the first mark’s
  2. half the first mark’s
  3. twice the first mark’s
  4. one quarter of the first mark’s
Show the answer

The wheel is a rigid system: every point on it turns through the same angle in the same time, so both marks have the same angular displacement (9.0 rad), the same ω and the same α.

  • Correct: the same as the first mark’s: Right: a rigid wheel turns as one piece, so every mark turns through the same angle.
  • half the first mark’s: Halving the distance from the axle changes how far the mark travels, not the angle it turns through.
  • twice the first mark’s: Being closer to the axle does not make a mark turn through more angle; the wheel turns as one piece.
  • one quarter of the first mark’s: No square law applies here: every mark on a rigid wheel turns through the same angle.

Graph

A ceiling fan switched on, then off

A sensor records the angular velocity of a ceiling fan, counterclockwise positive, as it is switched on, runs, and is switched off.

02468101214024681012141618Time (s)Angular velocity (rad/s)
Data table
Time (s)ω
00
412
1012
160
180

4. What is the fan’s angular acceleration from 0 to 4.0 s?

Type a number in rad/s^2.

Show the answer

α = Δω/Δt = (12.0 rad/s − 0)/(4.0 s) = 3.0 rad/s², the slope of the graph.

  • Answer: 3.0 rad/s^2

5. Through what angle does the fan turn from 0 to 16.0 s?

Type a number in rad.

Show the answer

The angle is the area under the ω-t graph: ½(4.0 s)(12.0 rad/s) + (6.0 s)(12.0 rad/s) + ½(6.0 s)(12.0 rad/s) = 24 + 72 + 36 = 132 rad.

  • Answer: 132 rad

6. Which describes the fan from 10.0 s to 16.0 s, with counterclockwise positive?

  1. Turning counterclockwise and slowing down
  2. Turning clockwise and speeding up
  3. Turning counterclockwise and speeding up
  4. At rest, since α is negative
Show the answer

ω is positive (counterclockwise) but decreasing toward zero, so the angular acceleration is negative, opposite to ω: the fan slows down until it stops at 16.0 s.

  • Correct: Turning counterclockwise and slowing down: Right: ω stays positive while its size falls, so α is negative.
  • Turning clockwise and speeding up: ω is still positive in this interval, so the fan has not reversed.
  • Turning counterclockwise and speeding up: ω is getting smaller, so the fan is slowing down.
  • At rest, since α is negative: A negative α means ω is decreasing, not that the fan has stopped; it stops at 16.0 s.

7. Counterclockwise is positive. A wheel has a negative angular velocity and a positive angular acceleration. What is it doing?

  1. Turning clockwise and slowing down
  2. Turning clockwise and speeding up
  3. Turning counterclockwise and slowing down
  4. Turning counterclockwise and speeding up
Show the answer

The sign of ω gives the direction of turning (negative: clockwise). When α has the opposite sign, ω moves toward zero, so the wheel slows, just as a car with v and a of opposite signs slows.

  • Correct: Turning clockwise and slowing down: Right: ω and α have opposite signs, so the size of ω is shrinking.
  • Turning clockwise and speeding up: Speeding up needs α with the same sign as ω.
  • Turning counterclockwise and slowing down: A negative ω means clockwise turning.
  • Turning counterclockwise and speeding up: The sign of ω, not α, gives the direction of turning.

8. A rigid wheel turns about a fixed axle. Which statement is true for the points on the wheel?

  1. Each point turns through the same angle in a given time
  2. Points farther from the axle turn through larger angles
  3. Each point moves the same distance in a given time
  4. Points near the axle have a larger angular velocity
Show the answer

A rigid system keeps its shape, so every point turns through the same angle in the same time: one θ, one ω and one α describe the whole wheel.

  • Correct: Each point turns through the same angle in a given time: Right: that is what it means for the wheel to be a rigid system.
  • Points farther from the axle turn through larger angles: Points farther out travel farther, but they turn through the same angle.
  • Each point moves the same distance in a given time: Points farther from the axle trace bigger circles, so they move farther.
  • Points near the axle have a larger angular velocity: Every point of a rigid wheel shares one angular velocity.

Part 9 · Summary

Summary

Rotation about a fixed axis is described by the angle θ in radians, the angular velocity ω = Δθ/Δt and the angular acceleration α = Δω/Δt, each with a sign for clockwise or counterclockwise. Every point of a rigid system shares them, and for constant α the three linear kinematic equations hold with θ, ω and α in place of x, v and a.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections