Topic 5.2 Connecting Linear and Rotational Motion: relate arc length, tangential velocity and tangential acceleration to the angular quantities through the radius, and distinguish tangential from centripetal acceleration.

What this topic is asking

The College Board (Topic 5.2) wants you to relate the linear quantities of a point on a rotating body, arc length, tangential velocity, tangential acceleration, to the angular quantities through the radius, and to distinguish tangential from centripetal acceleration. These relations are the bridge between the rotational kinematics of Topic 5.1 and the real motion of points on wheels, gears and rolling bodies.

Linking arc length and angle

These follow directly from $s = r\theta$ by differentiating with respect to time, since $r$ is fixed for a point on a rigid body: $\dfrac{ds}{dt} = r\dfrac{d\theta}{dt}$ gives $v = r\omega$, and differentiating again gives $a_t = r\alpha$. The tangential speed and acceleration are the actual linear speed and the along-the-path acceleration of the point. The radius is the conversion factor: the farther a point is from the axis, the faster it moves and the larger its tangential acceleration, even though it shares the body's angular velocity and angular acceleration.

Same angular motion, different linear motion

A key consequence: on a rigid body, every point has the same $\omega$ and $\alpha$ (the body turns as a whole), but the linear speed $v = r\omega$ and tangential acceleration $a_t = r\alpha$ depend on the radius. The rim of a wheel moves faster than a point near the hub; the tip of a fan blade moves faster than its base. This is why a long lever or a large gear amplifies linear speed at the expense of force, a theme that runs into torque and gearing.

Tangential versus centripetal acceleration

A point on a rotating body can have two accelerations at once, and they are perpendicular:

• Tangential acceleration $a_t = r\alpha$, along the direction of motion, which changes the speed. It is nonzero only when the rotation is speeding up or slowing down ($\alpha \ne 0$).
• Centripetal acceleration $a_c = \dfrac{v^2}{r} = \omega^2 r$, directed toward the axis, which changes the direction of the velocity. It is present whenever the point moves in a circle, even at constant angular speed.

Because they are perpendicular, the total acceleration has magnitude $a = \sqrt{a_t^2 + a_c^2}$. In uniform rotation ($\alpha = 0$) only the centripetal part remains, recovering the circular-motion result of Unit 2; when the rotation accelerates, both contribute.

Try this

Q1. A wheel of radius $0.40$ m turns at $5.0$ rad/s. Calculate the linear speed of a point on its rim. [2 points]

• Cue. $v = r\omega = (0.40)(5.0) = 2.0$ m/s.

Q2. Explain why two points at different radii on a rigid rotating disk have the same angular velocity but different linear speeds. [2 points]

• Cue. A rigid body turns as a whole, so $\omega$ is shared; but $v = r\omega$ scales with the radius, so the outer point is faster.