hello can you please type steps with formulas and numbers? there are similar ans

Question

hello can you please type steps with formulas and numbers? there are similar answers on here but i dont understand where the numbers come from or what formulas they are using. thank you

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 = 0.178 kg and moves at v = 4.72 m/s. The circular path has a radius of R = 0.9 m

3) What is the magnitude of the tension in the string when the ball is at the top of the circle?
?N
4) What is the minimum velocity so the string will not go slack as the ball moves around the circle?

Explanation / Answer

--->velocity is constant, this means the centripetal force is constant.
Fc = m * v^2 / r

= ( 0.178 * 4.72^2)/0.9

= 4.40 N 

---->The tension in the rope, the weight of the ball, and the centripetal force are all vertical forces. 

Tension = weight + Fc

Weight = 0.178 * 9.8 = 1.744 N
Tension = 1.744 + 4.40 N 

= 6.144( max tension)

3)

When the ball is at the top of the circle  the weight of the ball is actually helping to keep the ball moving at a constant speed. So the tension at the top of the circle is the minimum tension.

Tension = Weight – Fc

= 4. 40 – 1.744

=2.656 

4) 

As the ball falls, its kinetic energy increases because its potential energy is decreasing.  The distance that the ball falls and rises is the diameter of the circle. 

PE at top = mass * g * diameter

= 0.178 * 9.8 * 2 * 0.9 

Set KE at bottom = 0.178 * 9.8 * 2 * 0.9 

½ * 0.178 * v^2 = 0.178 * 9.8 * 2 * 0.9 

 v =35.28

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