Question

A 300g rock tied to a string is rotated in a vertical circle (up and down, so gravity is important) with radius 0.5m. Model the rock as a point particle. If a torque of 2Nm is applied for 3s, what is the rotation rate of the rock? What is the tension in the string at the top of the circle?

Answer 1

here,

radius of path, r = 0.5 m

mass of rock, m = 300 g = 0.3 kg

time, t = 3 s

torque, t = 2 N.m

since translational torque is:
t = Force * distance
t = Tension * radius of path
2 = Tension * 0.5
tension in string, T = 4 N

now,
rotational torque = tranlational torque
I*alpha = T*r (I is moment of inertia of system)

angular acceleration, alpha = tension * r / I
angular acceleration, alpha = T *r/mr^2
angular acceleration, alpha = 4 *0.5/0.3*0.5^2
angular acceleration, alpha = 26.667 rad/s^2

Since,
alpha = w^2*r
rotation rate, w = sqrt(alpha/r)
rotation rate, w = sqrt(26.667/0.5)
rotation rate, w = 7.303 rad/s