Questions: Rigid Body Kinematics — Fixed-Axis Rotation

Question 1

A wheel spins at a constant angular velocity (α = 0). What is the acceleration of a point on the rim?

• A. Zero — since angular acceleration is zero, no linear acceleration exists
• B. Only tangential acceleration, directed along the rim, because the speed is changing
• C. Only centripetal acceleration, directed toward the rotation axis, with magnitude ω²r
• D. Both centripetal and tangential acceleration, because the velocity vector is always changing

Answer: (C)

At constant angular velocity, α = 0, so tangential acceleration a_t = αr = 0. However, the point is moving in a circle — its velocity direction is constantly changing even though its speed is constant. This change in direction requires centripetal (normal) acceleration a_n = ω²r, directed radially inward toward the axis. This is the most common misconception: students assume 'constant speed' means 'no acceleration,' forgetting that acceleration is a vector and that changing direction counts. A point on a spinning wheel at constant ω has nonzero centripetal acceleration at every instant.

Question 2

A point P is located at radius r = 0.4 m on a rotating disk with ω = 5 rad/s and α = 3 rad/s². The tangential and centripetal (normal) accelerations of P are:

• A. a_t = 3 m/s², a_n = 5 m/s² — using a = αr and a = ωr directly
• B. a_t = 1.2 m/s², a_n = 10 m/s² — using a_t = αr and a_n = ω²r
• C. a_t = 0.6 m/s², a_n = 2 m/s² — dividing ω and α by r
• D. a_t = 1.2 m/s², a_n = 5 m/s² — using a_t = αr and a_n = ωr

Answer: (B)

Tangential acceleration a_t = αr = 3 × 0.4 = 1.2 m/s². Centripetal acceleration a_n = ω²r = 25 × 0.4 = 10 m/s². Note that centripetal acceleration uses ω² (not ω) multiplied by r. Option D is a common error: using ωr for centripetal acceleration instead of ω²r. Option A uses α and ω directly without multiplying by r. The total acceleration magnitude is √(a_t² + a_n²) = √(1.44 + 100) ≈ 10.1 m/s², showing that centripetal acceleration dominates at moderate angular speeds.

Question 3

A rigid body rotating at constant angular velocity (α = 0) has zero acceleration at most point.

• T. True
• F. False

Answer: False

Every point on a rotating body (except points on the axis itself) has centripetal acceleration a_n = ω²r directed toward the rotation axis, regardless of whether angular acceleration is zero. Constant angular velocity means zero tangential acceleration (a_t = αr = 0) — the speed of each point is not changing — but the direction of the velocity vector is continuously changing as the point travels in a circle. This change in direction is centripetal acceleration. Only when ω = 0 (body not rotating) does a_n also vanish. This is a critical point: 'no angular acceleration' ≠ 'no linear acceleration.'

Question 4

The centripetal acceleration of a point P at radius r on a rotating rigid body is directed toward the rotation axis and has magnitude ω²r.

• T. True
• F. False

Answer: True

This is the correct formula and direction for centripetal acceleration in fixed-axis rotation. The centripetal acceleration keeps point P moving in a circle by continuously redirecting its velocity vector toward the center (the rotation axis). Its magnitude is ω²r — note it depends on ω squared, so even modest angular speeds produce significant centripetal accelerations, especially at large r. This is distinct from tangential acceleration a_t = αr, which changes the speed and points along the arc (tangent to the circle). Both components are always perpendicular to each other.

Question 5

Why does a point on a spinning wheel have centripetal acceleration even when the wheel rotates at constant speed? What causes this acceleration, and in what direction does it point?

Model answer: Centripetal acceleration arises because the velocity of a point on the rim is constantly changing direction, even when its magnitude (speed) is constant. Velocity is a vector — any change in direction constitutes acceleration, even without a change in speed. As the point travels in a circle, its velocity vector must continuously rotate to remain tangent to the circular path. The rate of change of that velocity vector points radially inward, toward the rotation axis, with magnitude ω²r. This inward acceleration is centripetal acceleration — it is the acceleration required to maintain circular motion, not a consequence of speeding up.

A common intuition failure is equating 'acceleration' with 'speeding up or slowing down.' In rectilinear motion, acceleration does change speed. But in circular motion, even at constant speed, the direction of motion changes at every instant, and Newton's second law requires a net force (and therefore acceleration) to produce any change in velocity, including a change in direction. The centripetal acceleration a_n = ω²r is the vector pointing from point P toward the axis that accounts for this continuous direction change. Forgetting this component leads to errors in any dynamics problem involving rotating components.