Need Physics Help!?

Question

I am stuck on this problem and would be really greatful with any help on it. Thanks a bunch!

A carousel rotates at the rate of 0.15 rev/s with an 80-kg man standing at a point 2.0 m from the axis of rotation.

(a) What is the new angular speed when the man walks to a point 0.7 m from the center? Assume that the carousel is a solid 25-kg cylinder of radius 2.0 m. Answer is in rad/s

(b) Calculate the change in kinetic energy due to this movement. Answer is in Joules

Answer 1

This problem requires you to use the conservation of angular momentum, and then calculate the change in moment of inertia. Basically, moment of inertia is the rotational equivalent of mass. Moment of Inertia is defined as:

Volume in tegral of ( r² dm )

(I have no good way of writing this - check the Wikipedia article below)

Next, the Conservation of Angular Momentum can be written as:

I * w1 = I * w2

(a) To solve this problem, first, note that both the carosel and the man have the same rotational speed of .15 rev/s. Second, assume the man is a point mass. Then, the moment of inertia for the system can be calculated as:

I-carosel = (1/2) * M * R² (defined for a cylinder)

I-man = m * r1² (initial)

I-man = m * r2² (final)

The net moment of inertia for the initial sysem is simply:

I-initial = [(1/2) * M * R²] + [m * r1²]

I-initial = [(1/2)(25 kg)(2 m)²]+[80 kg * (2 m)²]

I-initial = 370 kg-m²

The net moment of inertia for the final sysem is simply:

I-final = [(1/2) * M * R²] + [m * r2²]

I-final = [(1/2)(25 kg)(2 m)²]+[80 kg * (.7 m)²]

I-final = 89.2 kg-m²

Using the conservation of angular momentum:

I-initial * w1 = I-final * w2

Re-arranging, we can solve for w2:

w2 = (I-initial / I-final) * w1

w2 = (370 / 89.2) * (.15 rev/s * 2 * pi rad/rev)

w2 = 3.91 rad/s

(b) The kinetic energy of a rotating body is given as:

KE = (1/2)*I*w²

Thus, the change in kinetic energy is:

KE = (1/2) ( I-final * w2² - I-initial * w1²)

KE = (1/2) ( 89.2 kg-m² * (3.91)² - 370 * (.15 rev/s * 2 * pi rad/rev)²)

KE = 517.3 J

That's it! :)

Answer 2

Angular momentum must be conserved. Angular momentum is I*w, where I is the moment of inertia and w the angular velocity. For the man, the moment of inertia is m*r^2, where m is his mass, and r his distance from the axis of rotation. The total moment of intertia is the sum of the man's plus that of the carousel. The moment of inertia of a solid cylinder is .5*mc*rc^2 where mc is the cylinder mass and rc is the cylinder radius. Add the moments of the cylinder and man both before and after the move. The angular speed will change inversely as the fractional change in moments:

EDIT: (note correction)

w(new) = I1/I2 * w0; (w0 is .15rev/sec = .15*2*π rad/sec, I1 be moment before move and I2 after.)

The kinetic energy of the rotating system is E=.5*I*w^2. Using the values of I and w before and after, compute the change.