Questions: Rigid Body Rotation: Angular Velocity and Acceleration

Question 1

A helicopter rotor blade is 6 meters long and spins at ω = 50 rad/s. A rivet 1 meter from the hub and the blade tip (6 meters from the hub) are compared. What is the ratio of the tip's linear speed to the rivet's linear speed?

• A. 1:1 — all points on a rigid body share the same angular velocity, so they move at the same speed
• B. √6:1 — linear speed scales with the square root of radius in rigid body rotation
• C. 6:1 — linear speed v = ωr scales linearly with radius, so the tip moves 6 times faster
• D. 36:1 — centripetal acceleration scales with r, and this ratio applies to speed as well

Answer: (C)

In rigid body rotation, all points share the same ω, but linear speed depends on radius: v = ωr. The rivet at r = 1m has speed v = 50(1) = 50 m/s; the tip at r = 6m has speed v = 50(6) = 300 m/s — six times faster. This is why helicopter rotor tips can approach or exceed the speed of sound while the hub barely moves. The shared ω is the rotational quantity that is uniform; the linear speed is the translational quantity that varies with radius. Option D confuses centripetal acceleration (which scales with ω²r) with speed.

Question 2

Two points on a spinning rigid disk — one at radius r and one at radius 4r — are compared as the disk undergoes angular acceleration α. How do their tangential accelerations compare?

• A. Both points have the same tangential acceleration because they share the same α
• B. The outer point has 4 times greater tangential acceleration because aₜ = αr scales with radius
• C. The outer point has 16 times greater tangential acceleration because acceleration scales with r²
• D. The inner point has greater tangential acceleration because it is closer to the source of rotation

Answer: (B)

Tangential acceleration is aₜ = αr. The point at radius 4r has tangential acceleration α(4r) = 4αr — four times greater than the point at radius r (which has aₜ = αr). Although both points share the same α, the linear effect of that angular acceleration grows with distance from the axis. This is why the outer edge of a spinning disk accelerates (in the linear sense) much more rapidly than the center when torque is applied. Note that centripetal acceleration aₙ = ω²r would give a 4:1 ratio as well — but only because both scale linearly with r.

Question 3

In a rotating rigid body, all points share the same angular velocity ω, which means a point at radius 2r has exactly twice the linear speed of a point at radius r.

• T. True
• F. False

Answer: True

v = ωr means linear speed is directly proportional to radius, with ω as the constant of proportionality. A point at 2r has v = ω(2r) = 2ωr — exactly twice the speed of the point at r where v = ωr. This linear scaling is a direct consequence of what 'rigid' means: all parts rotate together at the same ω, so the linear speed at any point is just ω times the distance from the axis. This relationship is fundamental to analyzing gears, pulleys, and any system where rotational and linear motions are coupled.

Question 4

Because angular acceleration α is the same for nearly every point in a rigid body, the tangential acceleration of most points in the body is also the same.

• T. True
• F. False

Answer: False

Angular acceleration α is uniform across the rigid body (all points rotate together), but tangential acceleration aₜ = αr varies with radius. A point close to the axis has small aₜ; a point at the rim has much larger aₜ. This is the distinction between angular quantities (ω, α) which are uniform for a rigid body, and linear/tangential quantities (v, aₜ) which depend on the distance from the axis. The confusion arises from conflating angular and linear acceleration — α being shared does not mean aₜ is shared, any more than sharing ω means sharing v.

Question 5

Explain why every point in a rigid rotating body shares the same angular velocity ω and angular acceleration α, even though their linear speeds and tangential accelerations are different. Why does the distinction between angular and linear quantities matter?

Model answer: A rigid body rotates as a single unit — no part deforms or moves relative to any other part. For this to hold, every point must sweep through the same angle in the same time interval, which means every point has the same angular velocity ω and the same rate of change of angular velocity α. If different parts had different ω values, the body would be twisting or deforming rather than rotating rigidly. However, linear speed v = ωr and tangential acceleration aₜ = αr convert these shared angular quantities into linear terms by scaling with radius. The same ω produces different linear speeds at different radii because speed measures how much distance is covered per unit time — and points farther from the axis trace longer arcs in the same time. The distinction matters because dynamics (torques, moments of inertia) operate in angular terms, while kinematics of individual points (velocities needed for stress analysis, gear teeth speeds) operate in linear terms.

This angular-vs-linear distinction is the conceptual core of rotational kinematics. Engineers need both: angular quantities to analyze the rotation as a whole (torque = Iα), and linear quantities to analyze individual points (stress in a rotor blade, speed of a gear tooth). The conversion v = ωr and aₜ = αr is the bridge between these two descriptions. Without understanding why angular quantities are uniform while linear quantities vary, students make errors both in setting up rotational dynamics problems and in interpreting the physical meaning of results.