A car on a track of radius R goes around in a circle at speed v. If R doubles without changing v, what happens to the required centripetal force?

A car on a track of radius R goes around in a circle at speed v. If R doubles without changing v, what happens to the required centripetal force?

• A. F_c doubles when R doubles
• B. F_c halves when R doubles
• C. F_c remains unchanged when R doubles
• D. F_c becomes zero when R doubles

Answer: (B)

For uniform circular motion, the centripetal force needed to keep a car moving in a circle is F_c = m v^2 / r. This means the required inward force is inversely proportional to the radius when speed is fixed. If the radius doubles while the speed stays the same, the factor in the denominator doubles, so the inward force becomes half as large. Intuitively, a larger circle with the same speed requires less inward pull to keep turning at the same rate. So the centripetal force is halved when the radius doubles. This doesn’t require more force, isn’t unchanged, and isn’t zero—the force scales with 1/r for constant v.

For uniform circular motion, the centripetal force needed to keep a car moving in a circle is F_c = m v^2 / r. This means the required inward force is inversely proportional to the radius when speed is fixed.

If the radius doubles while the speed stays the same, the factor in the denominator doubles, so the inward force becomes half as large. Intuitively, a larger circle with the same speed requires less inward pull to keep turning at the same rate.

So the centripetal force is halved when the radius doubles. This doesn’t require more force, isn’t unchanged, and isn’t zero—the force scales with 1/r for constant v.