Question

a wheel spins with an initial angular velocity of 18 rad/s in the clockwise direction and has a constant angular acceleration. after 3 seconds the wheel is spinning at 6 rad/s in the counterclockwise direction. what is the magnitude and direction of the angular acceleration?

Explanation:

To solve for the angular acceleration, we use the rotational kinematic equation:
$$\omega = \omega_0 + \alpha t$$
where:

• $\omega$ = final angular velocity,
• $\omega_0$ = initial angular velocity,
• $\alpha$ = angular acceleration,
• $t$ = time.

Step 1: Define the sign convention

Let counterclockwise (CCW) be positive and clockwise (CW) be negative.

• Initial angular velocity: $\omega_0 = -18 \, \text{rad/s}$ (CW is negative).
• Final angular velocity: $\omega = +6 \, \text{rad/s}$ (CCW is positive).
• Time: $t = 3 \, \text{s}$.

Step 2: Rearrange the kinematic equation to solve for $\alpha$

From $\omega = \omega_0 + \alpha t$, solve for $\alpha$:
$$\alpha = \frac{\omega - \omega_0}{t}$$

Step 3: Substitute the values

Substitute $\omega = 6$, $\omega_0 = -18$, and $t = 3$:
$$\alpha = \frac{6 - (-18)}{3} = \frac{6 + 18}{3} = \frac{24}{3} = 8 \, \text{rad/s}^2$$

Step 4: Determine the direction

The angular acceleration $\alpha = +8 \, \text{rad/s}^2$, which is positive. Since we defined CCW as positive, the angular acceleration is in the counterclockwise direction.

Answer:

The magnitude of the angular acceleration is $\boldsymbol{8 \, \text{rad/s}^2}$, and the direction is counterclockwise (CCW).