Question

Starting from rest, a wheel has constant acceleration $\mathbf{\alpha }\mathbf{=}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\frac{\mathbf{rad}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{}}$. During a certain role="math" localid="1660898447415" $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}\mathbf{}$interval, it turns through$\mathbf{120}\mathbf{}\mathbf{rad}$. How much time did it take to reach that $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}\mathbf{}$ interval?

Short answer

Wheel requires $8.0\text{s}$ time to reach that interval.

Step-by-step solution

Listing the given quantities

The angular acceleration of a wheel,$\mathrm{\alpha }=3.0\frac{\text{rad}}{{\text{s}}^2}$

The time interval of$120\mathrm{rad}$ displacement is,$\mathrm{t}=4.0\mathrm{s}$

The angular displacement is, $\mathrm{\theta }=120\text{rad}$

Understanding the kinematic equations

Use the kinematic equation for constant angular acceleration to calculate the initial angular velocity of the given interval. Use angular velocity formula to determinethefinal angular velocity asthewheel starts from rest.

Formula:

i) $\mathrm{\omega }={\mathrm{\omega }}_0+\mathrm{\alpha t}$

ii) $\mathrm{\theta }={\mathrm{\omega }}_0\mathrm{t}+\frac{1}{2}{\mathrm{\alpha t}}^2$

Calculate the angular velocity

The kinematic equation for angular motion is:

$\mathrm{\theta }={\mathrm{\omega }}_0\text{t}+\frac{1}{2}\mathrm{\alpha }{\text{t}}^2$

Substitute the value and solve as:

Solving for${\mathrm{\omega }}_0$:

${\mathrm{\omega }}_0=24\frac{\text{rad}}{\text{s}}$

Calculate the time required for the wheel to get the angular speed of  24 rads

Consider the expression for the angular speed as:

$\mathrm{\omega }={\mathrm{\omega }}_0+\mathrm{\alpha t}$

Solving for time, $\mathrm{t}=\mathrm{T}$

$\text{T}=\frac{\mathrm{\omega }-{\mathrm{\omega }}_0}{\mathrm{\alpha }}$

Substitute the values and solve as:

$\begin{array}{c}\text{T}=\frac{24-0}{3.0}\\ =8.0\text{s}\end{array}$

Time required for the wheel is $8.0\text{s}$.