What is the angular speed, and amount of work done?

Question

A woman whose mass is 54.7 kg stands at the rim of a horizontal turntable which has a moment of inertia of
338 kg m^2 about the axis of rotation and a radius of 2.26 m. The system is initially at rest and the turntable is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim in a clockwise direction (viewed from above) at a constant speed of 0.873 m/s relative to the ground.

What is its angular speed? Answer in units of rad/s.

How much work does the woman do to set the system into motion? Answer in units of J.

Answer 1

Even though the woman does work on the system the system will obey conservation of momentum.

So Lw+Lt=0
where Lw is the angular momentum of the woman and Lt is the angular momentum of the turntable.

since Lw=-Lt and
L=I*w
then the turntable rotates counter clockwise.

we know here angular speed to be
.873/2.26
=0.386 rad/sec

I will treat her moment of inertia as a point mass
54.7*2.26=338*w

where w is the angular speed of the turntable
w=54.7*2.26/338

=0.366 rad/sec counter clockwise

The energy can be calculated looking at the total rotational kinetic energy of the system.

Rotational energy is
.5*I*w^2
since this is a magnitude, simply add the turntable and woman energy together

.5*(54.7*2.26*0.386^2+
338*0.366^2)

=31.85 J

j