Question

A motor car is travelling at $60m/s$ on a circular road of radius $1200m$. It is increasing its speed at the rate of $4m/s$. The acceleration of the car is:
A. $3m/{{s}^{2}}$
B. $6m/{{s}^{2}}$
C. $5m/{{s}^{2}}$
D. $7m/{{s}^{2}}$

Answer 1

When a body is performing uniform circular motion then the tangential acceleration of the body is zero, but the radial acceleration of the body is non-zero.
When a body is performing accelerated circular motion then the tangential acceleration of the body is non-zero, also the radial acceleration of the body is non-zero.

Complete step by step solution:
If a body is in circular motion with velocity $v$ in circular path of radius $R$, then the radial acceleration of the body is given as
${{a}_{c}}=\dfrac{{{v}^{2}}}{R}$
It is given that the linear speed of the body is $60m/s$ in circular path of radius $1200m.$ The speed of the body is increasing at the rate of $4m/{{s}^{2}}$
The tangential acceleration of the body is given as,
${{a}_{t}}=\dfrac{dv}{dt}=4m/{{s}^{2}}$
Putting the values of linear speed and the radius of the circular path in the formula for radial acceleration, we get
$\begin{aligned} & {{a}_{r}}=\dfrac{{{\left( 60 \right)}^{2}}}{1200}m/{{s}^{2}} \\\ & =\dfrac{3600}{1200}m/{{s}^{2}} \\\ & =3m/{{s}^{2}} \end{aligned}$
The radial acceleration and the tangential acceleration are perpendicular to each other.
So, the angle between the radial acceleration and the tangential acceleration is $\theta =90{}^\circ$
As acceleration is a vector quantity. The net acceleration of the body is the resultant of the radial acceleration and the tangential acceleration,
$\begin{aligned} & a=\sqrt{{{\left( {{a}_{r}} \right)}^{2}}+{{\left( {{a}_{t}} \right)}^{2}}+2\left( {{a}_{r}} \right)\left( {{a}_{t}} \right)\cos \theta } \\\ & =\sqrt{{{\left( 3 \right)}^{2}}+{{\left( 4 \right)}^{2}}+2\left( 3 \right)\left( 4 \right)\cos 90{}^\circ } \\\ & =\sqrt{9+16}m/s \\\ & =\sqrt{25}m/s \\\ & =5m/s \end{aligned}$

Hence, the acceleration of the body is 5m/s.

Note: - The direction of tangential acceleration is along the tangent to the circular motion.
- The direction of radial acceleration is along the radius of the circular motion.