Question

(c) A string is wrapped many times around a frictionless pulley, which is a uniform disk of mass M. The string is connected to a hanging mass of massm. What is the speed of the hanging mass after it falls a distance h from rest?

(d) A ceiling fan accelerates uniformly at a rate of 3rad/s^2 from rest. How long does it take for the acceleration of a point 1.2 m from the center to have a magnitude of 6m/s^2

Answer 1

The net torque acting across the pulley is as follows Here, r is the radial distance and F is the force acting on the pulley Also, the torque in terms of the moment of inertia and the angular acceleration is Here. I is the moment of inertia and α is the angular acceleration The moment of inertia is equal to, And the angular acceleration is, al 4 Here, a is the acceleration Now, equate equation (1) and (2) as follows T-ma Thus, the tension in the pulley is equal to Ma Now, balancing the net force, mg-T=ma

Solve for a. 2mg M +2m Now, the velocity of the mass m is, 2mg h M +2m) Thus, the speed is, 2gh AR 1+ 2

The tangential acceleration is, 3 rad's (1.2 m) -3.6 m/s2 Assume the centripetal acceleration to be a =ω2 (1.2m) Thus, the net acceleration is, 36 (m/s(120') -(3.6 m/s')y 36 (m/s2). 1.44o' +12.96(ms2 )2 Solve for co 1440-36 (m/s')-12.96(m/s') o" 16 (m/s*). ω= 2 rad/s. From kinematics, 2 rad/s1- (3 m/s2)t Solve for t 2 rad/s2 = 0.67 s