Centripetal acceleration – choose the correct expression in terms of v, ω, and r For motion along a circular path of radius r with linear speed v and angular speed ω, which expression correctly gives the centripetal (radial) acceleration a_c?

Introduction / Context:
Centripetal acceleration quantifies the inward acceleration necessary to maintain circular motion. It is pivotal for analyzing curves in highways, rotor dynamics, and cable tensions in rotating machinery.

Given Data / Assumptions:

• Uniform circular motion of radius r.
• Linear speed v and angular speed ω are related by v = r * ω.
• Acceleration is directed toward the center (radially inward).

Concept / Approach:

Standard relations are a_c = v^2 / r and, equivalently using v = r * ω, a_c = r * ω^2. Both forms are correct; selecting one consistent expression answers the question. Dimensional analysis provides a quick validation.

Step-by-Step Solution:

Verification / Alternative check:

Units: v^2 / r has (m^2/s^2) / m = m/s^2. Likewise, r * ω^2 → m * (1/s^2) = m/s^2.

Why Other Options Are Wrong:

(b) r * ω has units of velocity, not acceleration. (c) and (d) have incorrect functional forms and units. (e) v * ω^2 gives m/s^3 units, not acceleration.

Common Pitfalls:

Dropping the square on v or ω; confusing tangential acceleration (if speed changes) with centripetal acceleration (directional change only).

Final Answer:

a_c = v^2 / r