A boy is swinging a toy on a piece of string in a vertical circle. The toy has a mass of 150 g and the radius of the circle is 0.8 m. a) He swings the toy with a linear velocity of 2 m/s. Will the toy move in a circle? Explain your answer. b) Another boy swings the toy with a linear velocity of 3.5 m/s. Work out the tension in the string at the top of the circle, at the bottom of the circle and halfway between the top and the bottom of the circle.

At the top of the circular motion, both weight and tension provides for centripetal force.

By Newton’s Second Law,
Fnet = ma
mg + T = mv^2/r (since a = v^2/r and weight = mg)

For toy to continue moving in circle at the top,

T > 0
mv^2/r - mg > 0
v >root (gr)

Hence, minimum speed toy must have is 2.80 m/s. Since linear velocity is lower than the minimum linear velocity, the toy will not move in circular motion.

b) Tension at top = mv^2/r - mg
= (0.15)(3.5)^2/0.8 - (0.15)(9.81)
= 0.825 N

Tension at bottom = mv^2/r + mg
= (0.15)(3.5)^2/0.8 + (0.15)(9.81)
= 3.77 N

In the middle, only Tension provides for centripetal force. Hence,
Tension = mv^2/r
= (0.15)(3.5)^2/0.8
= 2.30 N