Unit 5 · Topic 5.2 Beta

Linking Linear and Rotational Motion

A point a distance r from the axis of a turning rigid object travels s = rθ, moves at v = rω and has tangential acceleration a_T = rα, with angles in radians.

Practice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

On a merry-go-round the kids at the rim scream and the kids near the middle barely notice. Everyone goes around once in the same time, yet the ride at the edge is much faster. The link between turning and moving is one factor: the distance from the axis.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. One revolution is how many radians?

  1. 2π rad
  2. π rad
  3. 360 rad
  4. 1 rad
Show the answer

A full circle is 2π rad = 360°.

  • Correct: 2π rad:
  • π rad:
  • 360 rad:
  • 1 rad:

2. An object in uniform circular motion at speed v on radius r has centripetal acceleration

  1. v²/r
  2. v/r
  3. vr
  4. v²r
Show the answer

a_c = v²/r, toward the center.

  • Correct: v²/r:
  • v/r:
  • vr:
  • v²r:

Part 4 · See it

See it first

A merry-go-round seen from above turns counterclockwise about its center. Rider A sits near the rim and rider B halfway out on the same radius. Both turn through the same angle each second, so they share the same angular velocity ω. Their velocity arrows are perpendicular to the radius, and A’s arrow is twice as long as B’s, because A is twice as far from the axis: v = rω. A panel lists s = rθ, v = rω and a_T = rα, with angles in radians.
Riders A and B share one ω; A is twice as far from the axis, so A moves twice as fast: v = rω. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A point sits a distance r from the axisit moves on a circle of radius r as the object turns
  2. The object turns through θ (in radians)the point travels a distance s = rθ along its circle
  3. The object turns at ωthe point moves at speed v = rω, tangent to its circle
  4. ω changes at αthe point’s speed changes at a_T = rα
  5. A string or belt leaves the rim without slippingit moves with the rim: same distance, speed and acceleration

Part 6 · Key ideas

Key ideas

  • Every point of a rigid object shares θ, ω and α; how far and how fast a point moves depends on r.
  • s = rθ, v = rω, aT = rα, with angles in radians.
  • A point twice as far from the axis travels twice as far and twice as fast.
  • A string unwinding without slipping moves with the rim: its length, speed and acceleration are rθ, rω, rα.
  • Centripetal acceleration v²/r = rω² is a different quantity from the tangential rα.

Part 7 · Misconception

A common mistake

The wrong idea: Two gears or pulleys joined by a belt turn at the same angular velocity.

What actually happens: The belt makes their rim speeds equal: r₁ω₁ = r₂ω₂. The smaller pulley turns faster.

Part 8 · Check yourself

Check yourself

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

Diagram

valvereflectorWheel on a fixed axle (bikeupside down on a stand).Axle to valve: 0.33 mAxle to reflector: 0.20 mTurning at 12 rad/s, thenslows steadily to restin 6.0 s.

1. What is the speed of the valve while the wheel turns at 12 rad/s?

Type a number in m/s.

Show the answer

v = rω = (0.33 m)(12 rad/s) = 3.96 m/s, about 4.0 m/s. The radian drops out of the unit.

  • Answer: 4.0 m/s

2. Compared with the valve, the reflector has

  1. the same angular velocity and a smaller speed
  2. the same speed and a smaller angular velocity
  3. a larger angular velocity and the same speed
  4. the same angular velocity and the same speed
Show the answer

The wheel is rigid, so both share ω = 12 rad/s. Speed is v = rω, so the reflector (0.20 m) moves at 2.4 m/s and the valve (0.33 m) at about 4.0 m/s.

  • Correct: the same angular velocity and a smaller speed: Right: one ω for the whole wheel, and v = rω is smaller at smaller r.
  • the same speed and a smaller angular velocity: Every point on the wheel shares ω; it is the speed that depends on r.
  • a larger angular velocity and the same speed: Points closer to the axle do not turn faster; ω is shared.
  • the same angular velocity and the same speed: The reflector traces a smaller circle in the same time, so it moves more slowly.

3. While the wheel slows steadily to rest, what is the size of the valve’s tangential acceleration?

Type a number in m/s^2.

Show the answer

α = Δω/Δt = 12 rad/s ÷ 6.0 s = 2.0 rad/s² in size, so a_T = rα = (0.33 m)(2.0 rad/s²) = 0.66 m/s².

  • Answer: 0.66 m/s^2

4. If the wheel turned at 24 rad/s instead of 12 rad/s, the valve’s speed would be

  1. 2 times as large
  2. 4 times as large
  3. the same
  4. half as large
Show the answer

v = rω with r fixed at 0.33 m, so v ∝ ω: doubling ω doubles v (to about 7.9 m/s).

  • Correct: 2 times as large: Right: v = rω, so v is proportional to ω at fixed r.
  • 4 times as large: v = rω has no square; doubling ω doubles v.
  • the same: The valve stays at 0.33 m, so a larger ω means a larger speed.
  • half as large: A faster wheel moves the valve faster, not slower.

5. The tip of a ceiling-fan blade is 0.60 m from the axis and moves at 9.0 m/s. What is the fan’s angular velocity?

Type a number in rad/s.

Show the answer

ω = v/r = (9.0 m/s)/(0.60 m) = 15 rad/s.

  • Answer: 15 rad/s

6. A belt joins a small pulley (radius 0.10 m) to a large pulley (radius 0.30 m) without slipping. The small pulley turns at 30 rad/s. How fast does the large pulley turn?

  1. 10 rad/s
  2. 90 rad/s
  3. 30 rad/s
  4. 3.3 rad/s
Show the answer

The belt moves at one speed, which equals each rim’s speed: v = (0.10 m)(30 rad/s) = 3.0 m/s. The large pulley then turns at ω = v/r = 3.0/0.30 = 10 rad/s.

  • Correct: 10 rad/s: Right: the belt gives both rims the same speed, 3.0 m/s, so ω = 3.0/0.30.
  • 90 rad/s: This multiplies by the ratio of radii the wrong way; the larger pulley turns more slowly.
  • 30 rad/s: The pulleys are not one rigid body; the belt links their rim speeds, not their ω.
  • 3.3 rad/s: This divides by 9, as if speed scaled with r². Rim speed v = rω is linear in r.

7. A turntable speeds up with constant angular acceleration. A point at distance r from the axis has tangential acceleration a. A point at distance 3r has tangential acceleration

  1. 3a
  2. 9a
  3. a
  4. a/3
Show the answer

Every point shares α, and a_T = rα, so tripling r triples the tangential acceleration.

  • Correct: 3a: Right: a_T = rα with α shared, so a_T ∝ r.
  • 9a: a_T = rα has no square in r.
  • a: α is shared, but a_T also depends on r.
  • a/3: Points farther out speed up more, not less.

Part 9 · Summary

Summary

A point a distance r from the axis of a turning rigid object travels s = rθ, moves at v = rω and has tangential acceleration a_T = rα, with angles in radians. All points share θ, ω and α, but points farther out move farther and faster. A string or belt that does not slip moves with the rim it touches.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections