A wheel undergoes rotational motion according to the equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$ , where $\surd$ is in radians, c = 1.0 rad/s, and d = -0.50 rad/s³. The wheel's instantaneous angular acceleration at t = 2.0 s is A) -6.0 rad/s². B) -3 rad/s². C) 0 rad/s². D) 3 rad/s².

Question

Quizplus · Accepted Answer

The angular acceleration is found by differentiating the angular position equation twice with respect to time. The first derivative gives angular velocity: $\omega = \frac{d\phi}{dt} = c + 3dt^2$, and the second derivative gives angular acceleration: $\alpha = \frac{d^2\phi}{dt^2} = 6dt$. Substituting $d = -0.50 \, \text{rad/s}^3$ and $t = 2.0 \, \text{s}$, we get $\alpha = 6(-0.50)(2.0) = -6.0 \, \text{rad/s}^2$.

A Wheel Undergoes Rotational Motion According to the Equation ϕ(rad)=ct+dt3\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }ϕ(rad)=ct+dt3

A Wheel Undergoes Rotational Motion According to the Equation $\phi ( \mathrm { rad } ) = c t + d t ^ { 3 }$