What is the correct formula for centripetal acceleration a_c in terms of speed v and radius r?

What is the correct formula for centripetal acceleration a_c in terms of speed v and radius r?

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

Answer: (D)

Centripetal acceleration is the inward acceleration needed to keep an object moving in a circle at a constant speed. The faster you go, the more inward pull you need, and the tighter the circle (smaller radius), the larger that inward acceleration must be. The magnitude is a_c = v^2 / r, and it’s also equal to ω^2 r since ω = v / r. This makes intuitive sense: if you double the speed while keeping the same radius, the required inward acceleration goes up by a factor of four; if you increase the radius while keeping speed fixed, the needed inward acceleration decreases inversely with r. Checking units shows v^2 / r has units (m^2/s^2) / m = m/s^2, which matches acceleration. The other forms don’t fit the behavior or the units: v r grows with r and does not scale like acceleration; r / v has units of time; and v^2 r has incorrect units for acceleration.

Centripetal acceleration is the inward acceleration needed to keep an object moving in a circle at a constant speed. The faster you go, the more inward pull you need, and the tighter the circle (smaller radius), the larger that inward acceleration must be. The magnitude is a_c = v^2 / r, and it’s also equal to ω^2 r since ω = v / r.

This makes intuitive sense: if you double the speed while keeping the same radius, the required inward acceleration goes up by a factor of four; if you increase the radius while keeping speed fixed, the needed inward acceleration decreases inversely with r. Checking units shows v^2 / r has units (m^2/s^2) / m = m/s^2, which matches acceleration.

The other forms don’t fit the behavior or the units: v r grows with r and does not scale like acceleration; r / v has units of time; and v^2 r has incorrect units for acceleration.