Question

Which of the following statements is incorrect about uniform circular motion (where the symbols have their usual meanings)?

& \text{A) }\left| \dfrac{d\overrightarrow{r}}{dt} \right|=\text{constant} \\\ & \text{B) }\left| \dfrac{d\overrightarrow{v}}{dt} \right|\ne \text{Zero} \\\ & \text{C) }\dfrac{d\left| \overrightarrow{v} \right|}{dt}=\text{Zero} \\\ & \text{D) }\dfrac{d\overrightarrow{v}}{dt}=\text{Zero} \\\ \end{aligned}$$

Answer 1

We need to understand the different physical quantities involved in a uniform circular motion such as the acceleration, velocity and displacement in both linear motion as well as angular motion terms to find the solution for the given problem.

Complete step by step answer:
We know that the linear quantities of motion in a uniform circular motion changes their direction at every point on the circle. The linear velocity and linear acceleration are tangential to the point on the circle on that instant of time. But these changes are periodic in a uniform motion. The rate of change of linear velocity is the linear tangential acceleration which is a constant for the motion.

Now, we can consider each of the relations given in the problem to understand which among them is wrong.
$\text{A) }\left| \dfrac{d\overrightarrow{r}}{dt} \right|=cons\tan t$: The relation says that the magnitude of rate of change of the displacement of the particle undergoing uniform circular motion is always a constant. We know that the rate of change of linear displacement is the linear velocity. The magnitude of linear velocity in a uniform circular motion is always constant.
$\text{B) }\left| \dfrac{d\overrightarrow{v}}{dt} \right|\ne Zero$: We are given that the magnitude of rate of change of linear velocity is not zero. We know that the rate of change of linear velocity is the tangential acceleration. The acceleration has a definite value throughout the uniform motion. The relation is therefore correct.
$\text{C) }\dfrac{d\left| \overrightarrow{v} \right|}{dt}=zero$: The statement says that the rate of change of magnitude of the linear velocity is always zero. We know that the magnitude of linear velocity in a uniform circular motion is always a constant. Then, the derivative of this will be zero. The relation is correct.
$\text{D) }\dfrac{d\overrightarrow{v}}{dt}=zero$: We are given that the rate of change of linear velocity is always zero. From our experience, we understand that the linear velocity in a uniform circular motion changes its direction at every instant. This implies that the linear velocity is not a constant, so the derivative will not be zero. The given relation is wrong.
The required answer is given by option D.

Note:
The linear velocity is a vector quantity and therefore has both magnitude and direction. We should be careful in relating it with other quantities as the magnitude of this linear velocity is always constant and the direction changes with every instant of time.