## Related questions with answers

At t = 0 a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of $30.0 \text { rad} / \mathrm { s } ^ { 2 }$ until a circuit breaker trips at t = 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. Through what total angle did the wheel turn between t = 0 and the time it stopped?

Solution

VerifiedThe angular acceleration is constant. Thus, we will apply the equations of rotation with constant angular acceleration model.

In order to calculate the total angle, we will divide the entire interval from $t=0$ to the time the wheel stops into two intervals.

From $t=0$ to $t=2\;\mathrm{s}$:

$\theta-\theta_0=\dfrac{1}{2}(\omega_{0z}+\omega_z)t \quad (1)$

We will calculate $\omega_{z}$ first:

$\begin{align*} \omega_z&=\omega_{0z}+\alpha_zt\\\\\ \omega_z&=24+(30)(2)\\\\ \omega_z&= \boxed {84\;\mathrm{rad/s}} \end{align*}$

Substitute $\omega_z$ into Eq.1

$\begin{align*} \theta-\theta_0&=\dfrac{1}{2}(\omega_{0z}+\omega_z)t\\\\ \theta-\theta_0&= \dfrac{1}{2}(24+84)(2)\\\\ \theta-\theta_0&= \boxed{ 108\;\mathrm{rad}}\\ \end{align*}$

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