Question

A string is wrapped around a pulley with a radius of 2.0 cm and no appreciable friction in its axle. The pulley is initially not turning. A constant force of 50 N is applied to the string, which does not slip, causing the pulley to rotate and the string to unwind. If the string unwinds 1.2 m in 4.9 s, what is the moment of inertia of the pulley?

Answer 1

this is a problem dealing with torques, accelerations and moments of inertia

We will need to determine the torque acting on the pulley; since there is no diagram, I will have to assume that the force acts tangentially to the pulley and therefore generates a torque of

torque = F r = 50N x 0.02m = 1 Nm

the torque on an object generates an angular acceleration according to

torque = I alpha where alpha is the angular acceleration

recall that angular acceleration is related to linear acceleration by

alpha = a/R where R is the radius (0.02m) of the object

so if we can find a, the linear acceleration, we can find alpha and knowing torque, find moment of inertia

if the pulley unwinds 1.2 m in 4.9s, we find acceleration from

dist = 1/2 a t^2 or a = 2 d/t^2 = 2 x 1.2m/(4.9x)^2 = 0.1m/s/s

and alpha = a/R = 0.1m/s/s / 0.02m = 5rad/s/s

therefore, since

torque = moment of inertia x alpha

moment of inertia = torque/alpha = 1 Nm/5 rad/s/s = 0.2kgm^2