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Question: A motor car is moving with the speed of \(20m{s^{ - 1}}\) on a circular track of radius \(100\,m\). ...

A motor car is moving with the speed of 20ms−120m{s^{ - 1}} on a circular track of radius 100 m100\,m. If its speed is increasing at the rate of 3 ms−23\,m{s^{ - 2}}, the resultant acceleration is:
A. 3 ms−23\,m{s^{ - 2}}
B. 5 ms−25\,m{s^{ - 2}}
C. 2.5 ms−22.5\,m{s^{ - 2}}
D. 3.5 ms−23.5\,m{s^{ - 2}}

Explanation

Solution

Here we have to use the concept of centripetal acceleration and the formula of resultant acceleration to get the answer.
Centripetal acceleration is the property of a body travelling along a circular direction.
The acceleration is directed radially into the middle of the circle and has a magnitude equal to the square of speed of the body around the curve, divided by the distance from the centre of the circle to the moving body.

Complete step by step answer:
Tangential acceleration, at=3ms−2{a_t} = 3m{s^{ - 2}}
Velocity, v=20ms−1v = 20m{s^{ - 1}}
Radius, r=100 mr = 100\,m
Mathematically, the centripetal acceleration is given by:
ac=v2r =20×20100 =4 ms−2  {a_c} = \dfrac{{{v^2}}}{r} \\\ = \dfrac{{20 \times 20}}{{100}} \\\ = 4\,m{s^{ - 2}} \\\
So, the resultant acceleration is given by:
a=(at)2+(ac)2 =(3)2+(4)2 =5 ms−2  a = \sqrt {{{\left( {{a_t}} \right)}^2} + {{\left( {{a_c}} \right)}^2}} \\\ = \sqrt {{{\left( 3 \right)}^2} + {{\left( 4 \right)}^2}} \\\ = 5\,m{s^{ - 2}} \\\

So, the correct answer is “Option B”.

Additional Information:
Centripetal acceleration is caused due to centripetal force.
The portion of the force which is perpendicular to the velocity is the part which results in the centripetal force.
A centripetal force is a net force acting on an object to keep it travelling along a circular road. It is important to note that the centripetal force is not a fundamental force, but merely a name provided to the net force that induces the object to travel along a circular direction.
Just the direction of the velocity varies in a uniform circular motion, since the force is at the right angle to the movement. No work is performed as the speed is constant and thus the energy remains constant.

Note:
Here we have to pay attention to the units of the given physical quantities. If they are not in SI form we have to convert them otherwise our answer will be wrong.
The centripetal force is a circular motion, so the work done by the centripetal force is always zero.