# Rotation Problem/Total Kinetic Energy

1. Dec 1, 2008

### mburt3

1. The problem statement, all variables and given/known data
A spool of mass 8.67kg and radius 4.58m is unwound by a constant force 48.1N pulling on the massless rope wrapped around it. Assume the moment of inertia of the spool about O is 3/4mr^2 and it rolls w/o slipping. When the center of the spool has moved a distance d, find the total kinetic energy (In terms of F and d).

2. Relevant equations
Work= ke(final)- ke(initial)
I=3/4mr^2
torque=Fr=I(alpha)

3. The attempt at a solution
I'm feeling pretty lost, but I was thinking that if I just added the rotational energy + translational kinetic energy that that would give me the answer. Obviously it is not this easy and I have no idea how to put it in terms of force and distance.
T

2. Dec 2, 2008

### bsodmike

The most I can get is as follows, the angular velocity, as the force of the massless string is exerted on the circumference of the spool,

$$F=ma=\dfrac{48.1\;N}{8.67\;kg}=a=5.55\;ms^{-2}$$

From the definition, $$\tau=I\cdot\alpha$$,
$$\tau=\dfrac{3}{4}mr^{2}\times5.55=757\;Nm = Fr$$

Also note that from the definition of Nm, 757 Nm = 757 Joules.

Kinetic energy is,
$$F\cdot d=\dfrac{1}{2}mv^2$$

The circumference of the wheel is $$2\pi r$$, this needs to be factored into Fr to relate the rotation angle to the ultimate displacement of the center, O?

Hope you can make some progress from this...