Question

A tennis ball connected to a string is spun around in a vertical, circular path at a uniform speed. The ball has a mass m = 0.16 kg and moves at v = 5.33 m/s. The circular path has a radius of R = 1.13 m. What is the minimum velocity so the string will not go slack as the ball moves around the circle?

Answer 1

v=3.57m/s

Step-by-step explanation:

Hi, let's solve this!

The speed is minimal when the voltage T = 0N

This occurs when the balance is

T + m * g = m * v2 / r

Since the voltage is zero, then we cancel it and the equation is

m * g = m * v2 / r

We have the following data:

m = 0.16kg

g = 9.8m / s2

r = 1.13m

We cleared the speed and it is

`v=√(g*r)`

`v=\sqrt{9.8m/s^(2)* 1.13m}`

v=3.57m/s

Answer 2

We assign the variables: T as tension and x the angle of the string
The centripetal acceleration is expressed as v²/r=4.87²/0.9 and (0.163x4.87²)/0.9 = T+0.163gcosx, giving T=(0.163x4.87²)/0.9 – 0.163x9.8cosx.

(1)At the bottom of the circle x=π and T=(0.163x4.87²)/0.9 – .163*9.8cosπ=5.893N.
(2)Here x=π/2 and T=(0.163x4.87²)/0.9 – 0.163x9.8cosπ/2=4.295N.
(3)Here x=0 and T=(0.163x4.87²)/0.9 – 0.163x9.8cos0=2.698N.
(4)We have T=(0.163v²)/0.9 – 0.163x9.8cosx.

This minimum v is obtained when T=0 and v verifies (0.163xv²)/0.9 – 0.163x9.8=0, resulting to v=2.970 m/s.