Question

The circular motion of a particle with constant speed is
A. Periodic but no SHM
B. SHM but not periodic
C. Periodic and also SHM
D. Neither periodic nor SHM

Answer 1

For the motion of the particle to be periodic the time period of the motion should be constant. The equation for the time period of a body in a circular motion is $t = \dfrac{{2\pi r}}{v}$ and since $\pi$, $r$ and $v$ are constant, the time period is constant and the motion is periodic. For SHM $a \propto - x$ (acceleration should be antiparallel to the displacement of the particle) but for a body moving in a circular motion the acceleration is perpendicular to the displacement of the particle, hence it is not in simple harmonic motion.

Complete step by step answer:
In the circular motion of a particle with constant speed, a particle repeats its motion after a regular interval of time but does not oscillate about a fixed point.
Uniform circular motion describes the movement of an object travelling a circular path with constant speed. The one-dimensional projection of this motion can be described as a simple harmonic motion.
Periodic Motion – It is defined as the motion of an object continually repeating itself, such as repeatedly moving in a circular orbit. The Law of Inertia states that an object moves in a straight line unless acted upon by a force, so periodic motion requires force to create this special type of motion.
Let the particle be $P$
And, the radius of the circle be $r$
Then the time taken by the particle to complete one round of circle be $t$
Then time-period will be equal to
$t = \dfrac{{Circumference}}{{Angular{\text{ }}velocity}}$
We know that the circumference of a circle is equal to $2\pi r$ and let the angular velocity be $v$
Then,
$t = \dfrac{{2\pi r}}{v}$
We know that the radius of a circle $r$ is constant, such that the $2\pi$ will also be constant.
It is given that magnitude of the velocity (i.e. the speed) is also constant
Hence, $t$ will also be constant.
We know that periodic motion means the motion of an object continually repeats itself. As the motion of a particle will be repeated while moving in a circle. Hence, the moment of a particle is repeated. As we know the time period of a particle to complete rounds of the circle is also constant. Therefore, the motion of a particle is both repeated and constant. Therefore, it is a periodic motion.
Now, let us check whether it is simple harmonic or not,
Condition for SHM.
We know that for SHM acceleration is proportional to negative displacement.
$a \propto - x$
Where $x$ is displacement
$a$ is acceleration.
This means that the acceleration of the particle should be antiparallel to the displacement for the periodic motion to be a simple harmonic motion.
In the circular motion of a particle, the direction of acceleration and displacement will form an angle of $90^\circ$ . Therefore, the condition of SHM is not satisfied.
Therefore, in a circular motion of a particle with constant speed, the particle repeats its motion after a regular interval but does not oscillate about a fixed point. So, the motion of the particle is periodic but not simple harmonic.
Therefore, option A is correct.

Note:
In this type of question, one should know that in a circular motion, the speed can be constant but the velocity cannot remain constant as the direction of the velocity continuously keeps on changing as the particle moves over the circular path. Its magnitude may remain the same but direction changes. This concept will help to solve questions related to the circular motion of an object.