In the case of uniform circular motion of a body along a circular path, what is the nature of its acceleration vector throughout the motion?

Introduction / Context:
This question concerns uniform circular motion, a common topic in mechanics. In uniform circular motion, an object moves around a circle at constant speed. Even though the speed is constant, the velocity vector changes direction continuously, which implies the presence of acceleration. Knowing the direction and magnitude behaviour of this acceleration, called centripetal acceleration, is essential for understanding circular motion, orbits and rotating systems.

Given Data / Assumptions:

• The motion is uniform circular motion, meaning constant speed along a circular path.
• The path radius is constant.
• We are asked about the nature of acceleration, specifically its magnitude and direction.
• We neglect any tangential acceleration, since speed is constant.

Concept / Approach:
In uniform circular motion, the speed v of the body remains constant, but the direction of velocity changes continuously. Acceleration is defined as the rate of change of velocity. Because only the direction of velocity changes while magnitude is constant, the acceleration must be perpendicular to the velocity at every point. For circular motion, this acceleration points towards the center of the circle and is called centripetal acceleration. Its magnitude is given by a_c = v^2 / r, where r is the radius of the circle. Since v and r are constant in uniform circular motion, a_c has constant magnitude and is always directed radially inwards.

Step-by-Step Solution:
Step 1: Recognise that in uniform circular motion, speed v is constant but velocity vector changes direction. Step 2: Because velocity direction changes, there must be non zero acceleration; otherwise, velocity would remain unchanged. Step 3: The required force and acceleration to keep the body on a circular path are directed towards the center, known as centripetal force and centripetal acceleration. Step 4: The magnitude of centripetal acceleration is a_c = v^2 / r, where v and r are both constant for uniform circular motion. Step 5: Since v and r are constant, a_c has constant magnitude; its direction is always inward, pointing towards the center of the circle. Step 6: Therefore, acceleration is constant in magnitude and directed radially inwards throughout the motion.

Verification / Alternative check:
A vector diagram of velocity vectors at two neighbouring points on the circle shows that the change in velocity Δv points towards the center. Repeating this reasoning at different points confirms that acceleration always points to the center. Also, if the acceleration were zero, the object would travel in a straight line, not a circle. If the acceleration were tangential, the speed would change, contradicting uniform speed. Hence, centripetal acceleration must be purely radial and of constant magnitude in uniform circular motion.

Why Other Options Are Wrong:
Variable in magnitude but constant in direction is incorrect because the direction of acceleration in circular motion changes as the radius direction changes, though it always points towards the center. Variable in magnitude but tangential to the circle is wrong because in uniform circular motion, there is no tangential acceleration; tangential acceleration would change the speed. Zero acceleration is incorrect because a change in direction of velocity requires non zero acceleration; without acceleration, the path would be a straight line.

Common Pitfalls:
Students sometimes think that constant speed means zero acceleration, forgetting that acceleration depends on changes in velocity, including direction. Another mistake is to assume that acceleration must act along the direction of motion; in circular motion, acceleration is perpendicular to velocity. Remember: in uniform circular motion, acceleration is centripetal, always directed towards the center, and has constant magnitude v^2 / r.

Final Answer:
In uniform circular motion, the acceleration is constant in magnitude and directed radially inwards toward the center of the circle.