A particle is moving in a circle of radius 1 m with speed varying with time as v=(2 t) m / s. I

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A particle is moving in a circle of radius 1 m with speed...

A particle is moving in a circle of radius $1 \mathrm{~m}$ with speed varying with time as $v=(2 t) \mathrm{m} / \mathrm{s}$. In first $2 \mathrm{~s}$ (A) Distance traveled by the particle is $2 \mathrm{~m}$. (B) Displacement of the particle is ( $2 \sin 2) \mathrm{m}$. (C) Average speed of the particle is $1 \mathrm{~m} / \mathrm{s}$. (D) Average velocity of the particle is zero.

A particle is moving in a circle of radius $1 \mathrm{~m}$ with speed varying with time as $v=(2 t) \mathrm{m} / \mathrm{s}$. In first $2 \mathrm{~s}$
(A) Distance traveled by the particle is $2 \mathrm{~m}$.
(B) Displacement of the particle is ( $2 \sin 2) \mathrm{m}$.
(C) Average speed of the particle is $1 \mathrm{~m} / \mathrm{s}$.
(D) Average velocity of the particle is zero.

Key Concepts

Average Speed and Average Velocity

Average speed is defined as the total distance traveled divided by the time taken, making it a scalar quantity, whereas average velocity is the displacement divided by the time taken, making it a vector quantity. In contexts such as circular motion, even when a particle covers considerable distance along the path, the average velocity might be small or even zero if the final position is near or equal to the starting point.

Displacement vs Distance

Distance is a scalar quantity that represents the total length of the path traveled by a particle, while displacement is a vector quantity that indicates the straight-line distance from the initial to final position, including direction. In circular motion, particularly when the path does not coincide with a straight line, the displacement can be significantly less than the distance traveled, and its calculation often involves geometric reasoning.

Non-Uniform Circular Motion

Non-uniform circular motion involves a particle moving along a circular path with a speed that changes over time. This contrasts with uniform circular motion, where the speed is constant. The study of non-uniform motion incorporates both the changing magnitude of velocity (speed) and the constant change in direction, requiring analysis of tangential and radial components of acceleration.

Angular Displacement

Angular displacement is the measure of the angle through which a particle moves on a circular path, typically expressed in radians. It is directly related to the arc length travelled and the radius of the circle by the relation: arc length = radius × angular displacement. In problems involving circular motion, converting between linear and angular quantities is fundamental for understanding the motion.

Arc Length Calculation

Calculating the arc length involves integrating the instantaneous speed over time along the curved path. For a particle with variable speed, evaluating the total distance traveled requires accounting for the changing speed along the circular trajectory, which is a common application of integral calculus in kinematics.

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