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
