Suppose a wheel is initially rotating at 10.0 rad/s while undergoing constant angular acceleration reaching a speed of 30.0 rad/s after 36.0 seconds have elapsed.How long after the initial time has the wheel undergone half of the angular displacement that it will have gone through during the entire 36.0-second interval?
A)0.556 s
B)22.2 s
C)18.0 s
D)56.5 s

Question

Quizplus · Accepted Answer

The angular displacement ($\theta$) of an object undergoing constant angular acceleration can be found using the equation $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$, where $\omega_0$ is the initial angular velocity, $\alpha$ is the angular acceleration, and $t$ is the time. Given the initial ($\omega_0 = 10.0$ rad/s) and final ($\omega = 30.0$ rad/s) angular velocities, and the time interval ($t = 36.0$ s), we can find the angular acceleration ($\alpha$) using $\alpha = \frac{\Delta \omega}{\Delta t} = \frac{30.0 - 10.0}{36.0} = \frac{20.0}{36.0} = \frac{5}{9}$ rad/s$^2$.To find when the wheel has undergone half the angular displacement, we set the displacement at time $t$ to half the total displacement over 36 seconds. The total displacement can be calculated using the initial formula, substituting the values for $\omega_0$, $\alpha$, and $t = 36.0$ s. To find the time when the displacement is half of this total, we solve the equation $\frac{1}{2}(\omega_0 \cdot 36.0 + \frac{1}{2} \cdot \frac{5}{9} \cdot 36.0^2) = \omega_0 t + \frac{1}{2} \cdot \frac{5}{9} \cdot t^2$.However, a simpler approach to this problem uses the fact that for constant acceleration, the average velocity ($\omega_{avg}$) over a time period is the midpoint of the initial and final velocities. Thus, $\omega_{avg} = \frac{\omega_0 + \omega}{2} = \frac{10 + 30}{2} = 20$ rad/s. Since the displacement is proportional to the average velocity, and the average velocity up to the halfway point of displacement is the average of the initial velocity and the average velocity over the entire period, we find that the time to reach half the displacement is when the wheel's velocity is 20 rad/s, which occurs at the midpoint of the time interval because of the constant acceleration. Therefore, half the angular displacement occurs at half the time interval, which is $36.0 \, \text{s} / 2 = 18.0 \, \text{s}$, not at 22.2 s as initially indicated by the choice. The correct reasoning leads to 18.0 s, which is option C, not B as initially stated. This was a mistake in the explanation. The correct answer, based on the correct explanation, should be C, not B.

Suppose a Wheel Is Initially Rotating at 10