Question

As you drive down the road at 14 m/s, you press on the gas pedal and speed up with a uniform acceleration of 1.22 m/s^2 for 0.65 s. If the tires on your car have a radius of 33 cm, what is their angular displacement during this period of acceleration?
Answer should be in radians.

Answer 1

Concepts and reason

The concept required to solve this problem is linear and angular displacement. Initially calculate the linear displacement of the car using kinematic equation for linear displacement and then calculate the angular displacement using angular displacement formula.

Fundamentals

In order to calculate the linear displacement here used kinematic equation. The kinematic equations are a set of four equations that can be used to find the unknown information about an object's motion if other information is known.

The kinematic equation for displacement is:

$s = {v_0}t + \frac{1}{2}a{t^2}$

Where ${v_0}$ is the initial velocity, $t$ is the time for which the objects moved and $a$ is the acceleration of the object.

To find the angular displacement of tyres, divide the displacement by the radius of the tyres.

Angular displacement,

$\Delta \varphi = \frac{s}{r}$

Where $s$ is the displacement and $r$ is the radius of the tyre.

Calculate the displacement of the car with uniform acceleration, $1.22{\rm{ m/}}{{\rm{s}}^2}$ and initial velocity, $14{\rm{ m/s}}$ .

Substitute $14{\rm{ m/s}}$ for ${v_0}$ , $1.22{\rm{ m/}}{{\rm{s}}^2}$ for $a$ , and $0.65{\rm{ s}}$ for t in equation $s = {v_0}t + \frac{1}{2}a{t^2}$ as follows:

$\begin{array}{c}\\s = \left( {14{\rm{ m/s}} \times {\rm{0}}{\rm{.65 s}}} \right) + \frac{1}{2} \times 1.22{\rm{ m/}}{{\rm{s}}^2} \times {\left( {0.65{\rm{ s}}} \right)^2}\\\\ = 9.1{\rm{ m}} + {\rm{0}}{\rm{.258 m}}\\\\s{\rm{ }} = {\rm{9}}{\rm{.358 m}}\\\end{array}$

Angular displacement of the car with radius $33{\rm{ cm}}$ is,

$\Delta \varphi = \frac{s}{r}$

Substitute the values of $9.358{\rm{ m}}$ for s and $33{\rm{ cm}}$ for r in equation $\Delta \varphi = \frac{s}{r}$ as follows:

$\begin{array}{c}\\\Delta \varphi = \frac{{9.358{\rm{ m}}}}{{0.33{\rm{ m}}}}\\\\ = 28.36{\rm{ radians}}\\\end{array}$

Angular displacement of the car, $\Delta \varphi = 28.36{\rm{ radians}}$

Ans:

The Angular displacement of the car, $\Delta \varphi = 28.36{\rm{ radians}}$