You have a pulley 12.3 cm in diameter and with a mass of 2.2 kg. You get to wond

Question

You have a pulley 12.3 cm in diameter and with a mass of 2.2 kg. You get to wondering whether the pulley is uniform. That is, is the mass evenly distributed, or is it concentrated toward the center or the rim? To find out, you hang the pulley on a hook, wrap a string around it several times, and suspend your 1.4 kg physics book 1.4 m above the floor. With your stopwatch, you can find that it takes 0.59 s for your book to hit the floor. What is the moment of inertia if the pulley is uniform? kg m^2 What is the actual moment of inertia of the pulley based on your measurement? kg m^2 What do you conclude about the pulley? The pulley is uniform (within 1% of uniform) The mass of the pulley is concentrated toward the center The mass of the pulley is concentrated toward the rim

Explanation / Answer

here,
radius of puller, r = 12.3/2 = 6.15 cm = 0.0615 m
mass of pulley, m = 2.2 kg

mass of book, mb = 1.45 kg
height of book, h = 1.4 s
time taken, t = 0.59 s

Velocity, v = g/t = 9.81/0.59 = 16.627 m/s
Angular Velocity, w = v/r = 16.627/0.0615 = 270.358 rad/s

Moment of inertia of pulley about axis thrugh centre, I = 0.5 * m * r^2 = 0.5 * 2.2 * 0.0615^2
I = 0.00416 Kg.m^2

From Conservation of Energy,
Potential Energy of book = rotational KE of Pulley
mb*g*h = 0.5 * If*w^2

Solving for Moment of inertia , If = 2*mb*g*h/w^2 = 2* 1.45 * 9.81 * 1.4 / (270.358)^2
If = 0.0005448 kg.m^2

Part A:
I = 0.00416 kg.m^2

part b:
If = 0.0005448 kg.m^2

Part C:
The mass towards centre Since I > If

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