Which expression correctly expresses centripetal acceleration?

• A. a_c = v / r
• B. a_c = v^2 / r
• C. a_c = r / v
• D. a_c = ρ g h

Answer: (B)

Centripetal acceleration is the inward acceleration required to keep an object moving in a circle. Its magnitude depends on how fast you’re moving and how tightly you’re turning: faster speed increases the required inward pull, and a smaller radius means you must change direction more quickly.

For uniform circular motion, the speed v is constant, but the velocity direction changes continuously. The change in velocity over a short time Δt is directed toward the center and has magnitude about Δv ≈ v Δθ, where Δθ is the small angle the path subtends. In circular motion, Δθ ≈ s/r with arc length s = v Δt, so Δθ ≈ v Δt / r. Therefore the acceleration magnitude a_c ≈ Δv/Δt ≈ [v (v Δt / r)] / Δt = v^2 / r. This is the standard expression, and it also matches the alternative form a_c = ω^2 r, since v = ω r.

So the correct expression is a_c = v^2 / r, which has the right units (m/s^2) and increases with speed while decreasing with larger radius.