Connecting Linear and Rotational Motion — Worked Solutions

These solutions belong to Connecting Linear and Rotational Motion. Each step states why it is useful. Try the questions first, then use this page to check your reasoning.

Question 1

Two points are fixed to the same rigid disk at radii r and 2r. Which quantity is the same for both?

1. Rigid-body points sweep the same angle in the same time. Why: Rigidity keeps their angular positions locked relative to each other.

2. They share ω, while v_t = r|ω| and a_r = rω² change with radius. Why: The outer point has larger linear speed and radial acceleration, not a larger angular velocity.

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Question 2

A disk turns at constant nonzero angular velocity. Which statement is correct?

1. Constant ω means α = 0 and therefore a_t = rα = 0. Why: The speed is not changing.

2. Velocity still changes direction, requiring a_r = rω² inward. Why: Constant speed is compatible with nonzero acceleration on a curved path.

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Question 3

A point travels an arc 0.60 m at radius 0.20 m. What angular displacement magnitude produced it?

1. Use s = rΔθ with angle in radians. Why: Arc length, not straight displacement, is the length related to the angle.

2. Δθ = 0.60/0.20 = 3.0 rad. Why: The length units cancel, leaving the radian ratio.

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Question 4

A string unwinds without slipping from a 0.040 m-radius axle. It moves 0.60 m while the wheel turns. Find the angular displacement magnitude. If the string then has acceleration 0.20 m/s², find the angular acceleration magnitude.

1. Use the no-slip string relation s = rΔθ. Why: String travel matches axle surface arc only when contact does not slip.

2. Δθ = 0.60/0.040 = 15 rad. Why: The relevant radius is the axle where the string contacts the system.

3. Use a_t = rα to obtain |α| = 0.20/0.040 = 5.0 rad/s². Why: Tangential acceleration at that radius connects the string and rotational motions.

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Question 5

Two marks are fixed to a rigid disk at radii 0.10 m and 0.30 m. During the same steady-rotation interval, a student reports tangential speeds 0.40 m/s and 0.90 m/s. Each speed has a possible error of at most 0.03 m/s; take the radii as exact for this task. Are the measurements consistent with one angular speed? Show a comparison that decides, then propose a specific measurement check before claiming that the outer mark really has a different angular speed.

1. Infer angular speeds from v_t/r: the inner mark gives 4.0 rad/s and the outer gives 3.0 rad/s. Why: Points fixed to a rigid disk must share angular velocity even though their tangential speeds differ.

2. The allowed speed intervals give angular-speed intervals 3.7–4.3 rad/s and 2.9–3.1 rad/s. They do not overlap. Why: The stated speed errors are too small to reconcile the reported measurements with a single angular speed.

3. For example, recheck that both arc travels were measured over the same timestamps and that the camera views along the axis; time several complete turns independently. Why: Mismatched time intervals or perspective distortion can produce a systematic speed error. An independent turn-period measurement checks ω directly before discarding the rigid-disk model.

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