Prof. Armstrong sits on a (frictionless) chair that is spinning on its axis with an angular speed of 2.0 rev/s. His arms are outstretched and he holds a heavy weight in each hand. The moment of inertia of the Prof., the weights, and the chair is (in total) 10 kg·m². Armstrong pulls the weights closer towards his body so that the moment of inertia decreases to 3 kg·m².

a) What is the resulting angular speed of the chair?

b) What is the change in kinetic energy of the system?

c) Where did the increase in kinetic energy come from?

a) There are no (net) external torques acting on the system (which consists of the Prof., chair, and weights). Thus the total angular momentum of the system is conserved:

L = I = constant

We are given the initial and final moments of inertia, and the initial angular speed so we can get the final angular speed using

_final = I_{initialinitial}/ I_final

Let's convert the units of angular speed into rad/sec (this is almost always a good idea):

_initial = (2.0 rev/s)(2 rad/rev) = 4.0 rad/s

thus

_final = (10 kg·m²)(4 rad/s)/(3 kg·m²)

= 41.9 rad/s

or, if you prefer

= 6.67 rev/s

b) The initial kinetic energy is

K_initial = 1/2 I_{initial initial}²

= 1/2 (10 kg·m²)(4 rad/s)²

= 789 J

The final kinetic energy is, similarly,

K_final = 1/2 I_finalfinal²

= (1/2)

= 1/2 (3 kg·m²)(41.9 rad/s)²

= 2634 J

Thus the kinetic energy of the system has increased by

2634 J - 789 J = 1845 J

Note that we had to have changed units of angular speed to rad/s in order to get the energy in Joules.

c) The kinetic energy of the system increased; since the law of conservation of energy indicates that we cannot create energy out of nothing, it must have been originally in some other form. In this case, the Prof. did work on the weights to pull them inwards; thus, by doing this work, chemical energy stored initially in his arm muscles was converted into (rotational) kinetic energy; by pulling in the weights he literally pulled himself around faster.

Note: We did this as a demonstration in class, and so the answer should be in your lecture notes...

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