Question

A car travels around a flat circle on the ground, at a constant speed of$\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{/}\mathbf{s}$. At a certain instant the car has an acceleration ofrole="math" localid="1657010334562" $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$toward the east. What are its distance and direction from the center of the circle at that instant if it is traveling (a) clockwise around the circle and (b) counterclockwise around the circle?

Short answer

(a) Distance and direction from the centre for clockwise direction is 48m west.

(b) Distance and direction from the centre for counter clockwise direction is 48m west.

Step-by-step solution

The given data

1. Speed of car,$v=12.0\mathrm{m}/\mathrm{s}$
2. Acceleration of car,role="math" localid="1657010443923" $a=3.0\mathrm{m}/{\mathrm{s}}^2$

Understanding the concept of centripetal acceleration

If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in a uniform circular motion and has an acceleration $\overrightarrow{\mathrm{a}}$of constant magnitude. The direction of$\overrightarrow{\mathrm{a}}$ is toward the center of the circle or circular arc, and is said to be centripetal acceleration.

We can use the centripetal acceleration equation to find the distance of the car from the center of the circle and we know the acceleration of the car is the centripetal acceleration.

Formulae:

The acceleration of the body in centripetal motion,$a=\frac{v^2}{r}$ (i)

Where a is the centripetal acceleration, v is the speed, r is the radius of curvature

a) Calculation for distance and direction of car for clockwise motion

The left side figure is for a clockwise motion.

Hence acceleration and distance from the centre is same for both cases. And the value of distance is given using equation (i) as:

$r=\frac{v^2}{a}$

Substitute the $12\mathrm{m}/\mathrm{s}$ for v, and $3\mathrm{m}/{\mathrm{s}}^2$ for a in the above equation.

$$\begin{gathered} =\frac{{(12.0\mathrm{m}/\mathrm{s})}^2}{3.00\mathrm{m}/{\mathrm{s}}^2} \\ =48\mathrm{m} \end{gathered}$$

For clockwise motion, the distance of the car from the centre is 48 m and the direction of the car is towards the west.

b) Calculation for distance and direction of car for counter-clockwise motion

The right-side figure is for counter-clockwise motion. Hence acceleration and distance from the center are the same in both cases.

Hence, for counter-clockwise motion, the distance of the car from the centre is 48m and the direction of the car is towards the west.