Question

2.0 kg rock in a 0.80 ling sting is swung vertically in circles.
a) What does the speed of the rock to be so the string holds tight?
b) what is the pulling force if the speed is 5.0 m/s?

Answer 1

If the string remains taut at all positions, the minimum tangential speed of the rock has to generate a centripetal force equal to the weight of the rock, or

mg = mv²/r

Here
m=2 kg
r=0.80
g=9.8 m/s²

w=weight=mg=19.6 N
solve for v:
2(9.8)=2v²/0.8
v=√(196/25)
=2.8 m/s

The pulling force (tension in string) varies at different positions.
The tension when the rock is horizontal,
T=mv²/r
=2(5²)/0.8
=62.5 N
At the top of the circular motion,
tension
= T-mg
= 62.5-19.6 N
= 42.9 N

At the bottom, T adds to the weight.
Guess you can take it from here.