Uniform circular motion — direction of acceleration: A particle moves along a circle with uniform speed. At any instant, the direction of its acceleration vector is which of the following?

Introduction / Context:
Uniform circular motion (UCM) features constant speed but continuously changing velocity direction. This change in direction implies a nonzero acceleration known as centripetal acceleration. Recognizing its direction is vital in dynamics of rotating machinery, vehicle cornering, and orbital motion.

Given Data / Assumptions:

• Particle moves with constant speed on a circular path of radius r.
• Angular speed may be denoted ω; linear speed v = ω r.
• No tangential acceleration component (speed constant).

Concept / Approach:

The acceleration responsible for changing the velocity direction points toward the centre of curvature. Its magnitude is a_c = v^2 / r = ω^2 r. Because speed is constant, any tangential component would imply a change in speed, which is absent in UCM; hence acceleration must be purely radial and inward.

Step-by-Step Solution:

Verification / Alternative check:

Differentiating the velocity vector in polar coordinates shows a radial component −ω^2 r pointing toward the centre, with zero tangential component for constant ω.

Why Other Options Are Wrong:

(a) and (c) incorrectly suggest outward direction; (b) implies a tangential component which would change speed; (e) is arbitrary and not supported by UCM kinematics.

Common Pitfalls:

Confusing centripetal (inward) with centrifugal (a pseudo force in a rotating frame) which appears outward; assuming any tangential component exists when speed is constant.

Final Answer:

towards the centre (radially inward)