A wheel rotates about a horizontal axis. It has an outer radius R 1 = 18.8 cm an

Question

A wheel rotates about a horizontal axis. It has an outer radius R1 = 18.8 cm and mass M1 = 5.12 kg. Attached to it concentrically is a hub (shown cross-hatched) of radius R2 = 5.35 cm and mass M2 = 1.12 kg. The axle has negligible radius and mass. Both the wheel and the hub are solid and have uniform density. Suspended from a massless string that is wound around the hub is a weight of mass m = 530 g.

a) What is the acceleration of the hanging weight after it is released?

( in m/s2)

b) When the weight has fallen a distance 64.9 cm, what is the kinetic energy of the rotating wheel? (in J)

Explanation / Answer

For the problem , let us consider both wheel and hub to be uniform Disks.

I1 = moment of inertia of wheel = MR^2/2 = 0.09 Kg sq.m

I2 = moment of inertia of hub = mr^2/2 = 0.0016 Kg sq.m

I = I1 + I2 = 0.092 Kg sq.m

To get the acceleration of the mass:

mg - T = ma ................1

T*r = Ia1 .......................2

a is acceleration of the mass , T is tension in the string , r is radius of small hub , a1 is angular acceleration

a1 = a/r ...............3

a)Solving the 3 equations , we get a = 0.16 ms^-2

b)When mass has fallen s= 0.649 m , V^2 = 2*a*s

V = 0.46 m/s

w = Angular velocity = V/r = 8.52 Rad/s

KE of rotating wheel = (1/2)Iw^2 = 3.338 J

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