A sling-thrower puts a stone (0.260 kg) in the sling's pouch (0.0400 kg) and then begins to make the stone and pouch move in a vertical circle of radius 0.670 m. The cord between the pouch and the person's hand has negligible mass and will break when the tension in the cord is 37.0 N or more. Suppose the sling-thrower could gradually increase the speed of the stone.
(a) Will the breaking occur at the lowest point of the circle or at the highest point?
(b) At what speed of the stone will that breaking occur?

a. It will break at the lowest point b. 8.72 m/s

Explanation:

a. The forces in the sling at its highest point are tension and weight which equals the centripetal force, since they all act downwards. So,

-T - mg = -mv²/r

T + mg = mv²/r

So, T = mv²/r - mg. This is also the normal force on the sling At this point, the stone is loosing contact with the sling, So, T = 0 and g = v²/r

At the lowest point of the circle, the tension acts upwards. So,

T - mg = mv²/r

T = mv²/r + mg. Here, the tension is maximum and this tension equals the normal force acting on the sling. Thus, it will break at this point.

b. The breaking will occur if T ≥ 37.0 N

T - mg = mv²/r

v = √(Tr/m - gr)

  = √(37 N × 0.670 m/(0.260 kg + 0.0400 kg) - 9.8 m/s²× 0.670 m)

  = √(82.63 m²/s² - 6.566 m²/s²)

  = √76.064

  = 8.72 m/s

365.25 earth days are in the year of a light year