12.1.1 Brief Review of Rotational Kinematics

Rotation is an extension of circular motion, so we begin with a brief summary of Chapter 4. A review of 《 SECTIONS 4.4– 4.6 is highly recommended. FIGURE 12.2 shows a wheel rotating on an axle. Its angular velocity

ω = \frac{d θ}{d t}

(12.1)

is the rate at which the wheel rotates. The SI units of $ω$ are radians per second $(rad / s)$ , but revolutions per second $(rev / s)$ and revolutions per minute (rpm) are frequently used. Notice that all points have equal angular velocities, so we can refer to the angular velocity $ω$ of the wheel.

The figure illustrates a rotating circular solid with a dot labeled Axle at the center.

FIGURE 12.2 Full Alternative Text

If the wheel’s rotation is speeding up or slowing down, its angular acceleration is

α = \frac{d ω}{d t}

(12.2)

The units of angular acceleration are ${rad/s}^{2}$ . Angular acceleration is the rate at which the angular velocity $ω$ changes, just as the linear acceleration is the rate at which the linear velocity v changes. TABLE 12.1 summarizes the kinematic equations for rotation with constant angular acceleration.

TABLE 12.1 Rotational kinematics for constant angular acceleration

ω_{f} = ω_{i} + α Δ t

θ_{f} = θ_{i} + ω_{i} Δ t + \frac{1}{2} α {(Δ t)}^{2}

ω_{f}^{2} = ω_{i}^{2} + 2 α Δ θ

FIGURE 12.3 reminds you of the sign conventions for angular velocity and acceleration. They will be especially important in the present chapter. Be careful with the sign of $α$ . Just as with linear acceleration, positive and negative values of $α$ can’t be interpreted as simply “speeding up” and “slowing down.”

The figure consists of four rotating circular solids that illustrate the signs of angular velocity and acceleration.

FIGURE 12.3 Full Alternative Text

A point at distance r from the rotation axis has instantaneous velocity and acceleration, shown in Figure 12.2, given by

\begin{array}{l} v_{r} = 0 & a_{r} = \frac{v_{t}^{2}}{r} = ω^{2} r \\ v_{t} = r ω & a_{t} = r α \end{array}

(12.3)

The sign convention for $ω$ implies that $v_{t}$ and $a_{t}$ are positive if they point in the counter-clockwise (ccw) direction, negative if they point in the clockwise (cw) direction.

12.1.1 Brief Review of Rotational Kinematics

FIGURE 12.2 Rotational motion.

TABLE 12.1 Rotational kinematics for constant angular acceleration

FIGURE 12.3 The signs of angular velocity and angular acceleration.