You\'ve been given a pulley for your birthday. It\'s a fairly big pulley, 12 cm

Question

You've been given a pulley for your birthday. It's a fairly big pulley, 12 cm in diameter and with a mass of 2.0 kg. You wonder whether the pulley is uniform. That is, is the mass evenly distributed, or is it concentrated toward the center or near the rim? To find out, you hang the pulley on a hook, wrap a string around it several times, and suspend your 1.0 kg physics book 1.0 m above the floor. With your stopwatch, you find that it takes 0.71 s for your book to hit the floor. What can you conclude about the pulley?

Explanation / Answer

the moment of inertia of the disk is

I_disk = (1/2) m r2 = (1/2) * 2.0 * 0.062 =   0.0036 kg-m2

from the kinematic equation theacceleration of the book is

     d = (1/2) at2     

a = 2d / t2

= 2 * 1.0 / 0.712

=   3.9675 m/s2

final speed of the book is

    v = a t

= 3.9675 *0.71

= 2.817 m/s

Apply conservation of energy

initila PE of book = finalKE of pulley + final KE of book

          m gh =    (1/2) I 2   +   (1/2) m v2

m g h = (1/2) I (v/r)2 + (1/2) mv2

1.0 * 9.8 *1.0   = (1/2) I (2.817 / 0.06)2 + (1/2) * 1.00 * 2.8172

9.8 =    1102 I  +    3.967

             I =   0.00529    kg-m2

I gretaer than the of I_ disk hence the mass of yourpulley must be concentrated at the rim.

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