Centripetal force in circles
Circular motion needs a real force pointed at the center, and identifying which force plays that role is the whole problem.
Centripetal acceleration
An object in uniform circular motion is accelerating even at constant speed, because the direction of the velocity vector keeps changing. The acceleration points toward the center with magnitude v^2/r, growing with speed and shrinking with radius. Doubling the speed quadruples the required force.
a_c = \frac{v^2}{r}What provides it
There is no new kind of force called centripetal force; the word names a role. Tension pulls a swung ball toward your hand, gravity keeps satellites in orbit, and friction points toward the center for a car on a flat curve. Newton's second law applied radially reads: the sum of real forces toward the center equals m v^2/r.
\sum F_{\text{toward center}} = \frac{mv^2}{r}Worked example
A 0.50 kg ball is swung in a horizontal circle on a 1.5 m string at 6.0 m/s. The centripetal acceleration is 36/1.5 = 24 m/s^2, so the string tension must be 0.50 times 24 = 12 N. If the string can only hold 10 N, the ball cannot be swung that fast: at the breaking tension the maximum speed is sqrt(10(1.5)/0.50) = sqrt(30) = 5.5 m/s.
T = \frac{mv^2}{r} = \frac{(0.50)(36)}{1.5} = 12 \text{ N}Flashcard
Centripetal acceleration toward the center
Check your understanding
A car takes a flat curve of radius 50 m at 20 m/s. What minimum coefficient of static friction keeps it on the road? Use g = 9.8 m/s^2.