Question

A particle is performing UCM along a circle of radius $r$. In one half revolution, what is the distance travelled and the magnitude of its displacement?

Answer 1

In uniform circular motion (UCM) the speed of particles is constant. This means that the time taken to complete the first half of the revolution will be the same as the time taken to complete the second half of the revolution. Distance is a scalar quantity that refers to the total path covered while displacement is a vector quantity. Displacement is the shortest path covered by the body.

Complete step by step answer: We are given with the radius of the circle. We are required to find the distance travelled and the magnitude of displacement for the one half of the revolution.

Let’s find the distance travelled in one half of the revolution. In one half revolution, the particle will trace the semi-circular arc. The circumference of a circle $= 2\pi r$
Therefore, one half revolution will have half of circumference.
One half of circumference $= \dfrac{{2\pi r}}{2} = \pi r$.
Therefore, the distance travelled by the particle is $\pi r$ units.

Displacement is the shortest distance that a particle can travel. In this case the particle is travelling along the curve of semicircle. The shortest distance to travel from one end of a semicircle to another is by going through the diameter.

We know that the diameter of the circle is twice the radius. Therefore, the distance travelled will be equal to the diameter of the circle.
The diameter of circle with radius $r$ units is $2r$
Therefore, the displacement of the particle in one half revolution is $2r$ units.

The distance is $\pi r$ units and the displacement is $2r$ units.

Note: Remember that in UCM the speed of the particle is constant. Displacement is the shortest distance possible from source to destination. As displacement is a vector, it has a magnitude as well as direction.