Question

1. a ball has a mass of 0.6 kg. it is twirled in a circle of radius 7 m. the time for one revolution is 1.5 minutes.

a.) what is the speed of the ball?
b.) what is the centripetal acceleration of the ball?
c.) how much force is required to keep the ball in this circle at this speed?
d.) if the ball goes twice as fast, how much more force is required?

Explanation:

Step1: Convert time to seconds

$1.5 \text{ minutes} = 1.5 \times 60 = 90 \text{ s}$

Step2: Calculate circumference of circle

$C = 2\pi r = 2\pi \times 7 = 14\pi \text{ m}$

Step3: Find speed (part a)

Speed $v = \frac{\text{Distance}}{\text{Time}} = \frac{14\pi}{90} \approx 0.4887 \text{ m/s}$

Step4: Find centripetal acceleration (part b)

$a_c = \frac{v^2}{r} = \frac{(0.4887)^2}{7} \approx 0.0338 \text{ m/s}^2$

Step5: Find centripetal force (part c)

$F_c = m a_c = 0.6 \times 0.0338 \approx 0.0203 \text{ N}$

Step6: Force at double speed (part d)

New speed $v' = 2v$, so new force $F_c' = m \frac{(2v)^2}{r} = 4m \frac{v^2}{r} = 4F_c$
Additional force $= 4F_c - F_c = 3F_c = 3 \times 0.0203 \approx 0.0609 \text{ N}$

Answer:

a.) $\approx 0.49 \text{ m/s}$
b.) $\approx 0.034 \text{ m/s}^2$
c.) $\approx 0.020 \text{ N}$
d.) $\approx 0.061 \text{ N}$ (3 times the original force)