Merry-go-round with brick and sliding friction.

AbigailM

1. The problem statement, all variables and given/known data
The picture shows a rotating platform that serves as a playground
merry go round. The platform rotates on low-friction bearings about its center
axis. It has a radius of 2.0 meters, and its moment of inertia about the center
axis is 200 kg m2.

Suppose that the platform is given an initial rotation rate of 1.0 radians per
second. A small dense brick having a mass of 50 kg, initially at rest, is then
placed on the platform at a distance of 0.5 meters from the rotation axis. The
brick initially slides, but eventually stops sliding at a distance 1.0 m from the
axis. How many Joules of mechanical energy are converted to heat?

2. Relevant equations
[itex]I_{brick}=mr^{2}[/itex]

[itex]L_{i}=L_{f}=>I_{i}\omega_{i}=I_{f}\omega_{f}[/itex] (cons. of angular momentum)

[itex]ΔK=-fx +W_{ext}[/itex] (work-energy theorem)

3. The attempt at a solution
[itex]I_{brick i}=(50kg)(0.25m^{2})=12.5kg.m^{2}[/itex]

[itex]I_{brick f}=(50kg)(1m^{2})=50kg.m^{2}[/itex]

[itex]I_{i}=I_{disk i}+I_{brick i}=212.5kg.m^{2}[/itex]

[itex]I_{f}=I_{disk f}+I_{brick f}=250kg.m^{2}[/itex]

[itex]\omega_{f}=\frac{I_{i}}{I_{f}}\omega_{i}=0.85\frac{rad}{s}[/itex]

[itex]ΔK=-fx[/itex] (no external work, just friction)

[itex]\frac{1}{2}I_{i}\omega_{i}^{2}-\frac{1}{2}I_{f}\omega_{f}^{2}=15.94 J[/itex]

Just wondering if my solution method is looking ok. Thanks for the help!

TSny

The wording here might be open to interpretation. I'm thinking that the brick is initially at rest with respect to the earth when it is let go at 0.5 m from the axis. It is therefore slipping on the platform until friction finally brings it to rest relative to the platform at 1.0 m from the axis. If so, would the brick contribute to the initial angular momentum?