Question

a car with tires of radius $r_0$ is moving along a straight road. each tires angular displacement is given by the equation $\theta = at^4 + bt$, where $t$ is time and $a$ and $b$ and are constants with appropriate units. the tires do not slip on the roads surface. what is the cars acceleration as a function of time?

Explanation:

Step1: Find angular velocity (ω)

Angular velocity is the derivative of angular displacement (θ) with respect to time (t). Given \( \theta = At^4 + Bt \), we differentiate:
\( \omega = \frac{d\theta}{dt} = 4At^3 + B \)

Step2: Find angular acceleration (α)

Angular acceleration is the derivative of angular velocity with respect to time:
\( \alpha = \frac{d\omega}{dt} = 12At^2 \)

Step3: Relate linear acceleration (a) to angular acceleration (α)

For rolling without slipping, linear acceleration \( a = r_0 \alpha \) (where \( r_0 \) is tire radius). Substitute \( \alpha \):
\( a = r_0 \cdot 12At^2 = 12Ar_0t^2 \)

Answer:

The car’s acceleration as a function of time is \( \boldsymbol{12 A r_0 t^2} \) (or \( 12 r_0 A t^2 \)).