Question

$(1)$ : In uniform circular motion the kinetic energy of the body is constant.
$(2)$ : In uniform circular motion the tangential force is zero.
(A) Both $\left( 1 \right)$ and $\left( 2 \right)$ are true and $\left( 2 \right)$ is the correct explanation of $\left( 1 \right)$
(B) Both $\left( 1 \right)$ and $\left( 2 \right)$ are true and $\left( 2 \right)$ is not the correct explanation of $\left( 1 \right)$
(C) $\left( 1 \right)$ is true and $\left( 2 \right)$ is false
(D) $\left( 1 \right)$ is false and $\left( 2 \right)$ is true

Answer 1

Hint: Uniform circular motion is a type of motion in which the body follows a circular path with a constant or uniform speed. We apply the conditions of uniform circular motion in the formula of kinetic energy and tangential force to solve this problem.
Formula used: kinetic energy $k = \dfrac{{m{v^2}}}{2}$
Here
Kinetic energy is represented by $k$
Velocity is represented by $v$
Mass is represented by $m$
The second law of motion $F = ma$
Force is represented by $F$
Acceleration is represented by $a$

Complete Step by step solution:
$\left( 1 \right)$ in a uniform circular motion, the speed at which the body is traveling is constant. At any particular point on the circular path followed by the body, we can say that both speed and velocity are the same. Therefore velocity is also constant.
Using the formula of kinetic energy
 $k = \dfrac{{m{v^2}}}{2}$
Velocity in a circular motion is represented in terms of angular velocity and is equal to
$v = \omega r$
Here
Angular velocity is represented by $\omega $
The radius of the circular path is represented by $r$
Velocity is represented by $v$
The angular velocity is also constant in the case of uniform circular motion as velocity is constant.
Mass is a constant quality
Since both mass and angular velocity are constants and kinetic energy depends only on these two qualities, kinetic energy is constant.
$\left( 2 \right)$ Tangential force in a circular motion is the product of mass and tangential acceleration.
Acceleration is the change in initial and final velocities. The velocity in a uniform circular motion is constant therefore the change in velocity and tangential acceleration is zero.
$\Delta v = 0$
$ \Rightarrow a = m(\Delta v) = 0{\text{ }}$
Since tangential acceleration is zero.
Tangential force is equal to $F = ma = m \times 0 = 0$
Because the tangential acceleration is zero it implies that the velocity or angular velocity is constant. We know that constant velocity is the reason for kinetic energy to be constant. Hence $\left( 2 \right)$ is the correct explanation for $\left( 1 \right)$

Option (A) “Both $\left( 1 \right)$ and $\left( 2 \right)$ are true and $\left( 2 \right)$ is the correct explanation of $\left( 1 \right)$ “ is the correct answer.

Note: Students might get confused with the concept of tangential acceleration and radial acceleration. In a uniform circular motion, tangential acceleration is equal to zero but radial acceleration ( along the radius) is equal to the centripetal acceleration.