Question

A particle moving in a circular path, covers equal distances in equal intervals of time. Then the quantity associated with the particle that remains constant with time is

• A. displacement
• B. velocity
• C. speed
• D. acceleration
• E. linear momentum

Hint:

In uniform circular motion, speed is constant but velocity is not, because its direction changes continuously.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question describes a particle in uniform circular motion. The phrase "covers equal distances in equal intervals of time" is the definition of constant speed. We need to analyze the vector quantities associated with this motion to see which one, if any, remains constant.
Step 2: Detailed Explanation:
Let's analyze each quantity for a particle in uniform circular motion:

(C) Speed: The problem statement itself defines that the particle has constant speed. Speed is the magnitude of velocity, a scalar quantity. Since it covers equal distances in equal times, its speed is constant.

(B) Velocity: Velocity is a vector quantity, having both magnitude and direction. The magnitude of the velocity (which is the speed) is constant. However, since the particle is moving in a circular path, the direction of its motion is continuously changing (it's always tangent to the circle). Because the direction changes, the velocity vector is not constant.

(A) Displacement: Displacement is the straight-line vector from the starting point to the ending point. As the particle moves around the circle, its position vector changes, and so does its displacement vector from the origin (or any other fixed point). The displacement over one full revolution is zero, but it is not constant throughout the motion.

(D) Acceleration: In uniform circular motion, there is always a centripetal acceleration directed towards the center of the circle. This acceleration is responsible for continuously changing the direction of the velocity. The magnitude of this acceleration is constant (\( a_c = v^2/r \)), but its direction is always changing as it always points towards the center from the particle's current position. Since the direction changes, the acceleration vector is not constant.

(E) Linear Momentum: Linear momentum is given by \( \vec{p} = m\vec{v} \). Since the mass \( m \) is constant and the velocity vector \( \vec{v} \) is continuously changing direction, the linear momentum vector \( \vec{p} \) is also continuously changing direction. Thus, it is not constant.

Step 3: Final Answer:
The only quantity that remains constant for a particle in uniform circular motion is its speed. This corresponds to option (C).