Conservation of Angular Momentum of Turntable

In summary, the problem deals with conservation of angular momentum and involves a 60-kg woman standing on a horizontal turntable with a moment of inertia of 500 kgm^2 and a radius of 2.00 m. When the woman starts walking around the rim clockwise at a constant speed of 1.50 m/s relative to the Earth, the turntable rotates in the opposite direction with an angular speed of 0.360 rad/s. This is due to the conservation of angular momentum, where the woman's angular momentum is equal and opposite to that of the turntable.
  • #1
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[SOLVED] Conservation of Angular Momentum

Homework Statement



A 60-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 kgm[tex]^{}2[/tex] and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. In what direction and with what angular speed does the turntable rotate?

Homework Equations



L(initial) = L(final)
I [tex]\omega[/tex](initial) = I [tex]\omega[/tex](final)

The Attempt at a Solution



My book lists this as a conservation of angular momentum problem and gives an answer of 0.360 rad/s. However, I can't seem to figure out how to apply the conservation of angular momentum, and how to incorporate the momentum of the woman into the problem. Everything is initially at rest, then the woman starts walking and everything is in motion. From where to where do I examine momentum? I'm assuming that the woman has a linear momentum since she does not have a moment of inertia, but I'm not sure. Any help would be greatly appreciated!
 
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  • #2
The woman is walking around the center of rotation of the wheel. She does have a moment of inertia. Treat her as a point mass M at distance R from the axis. Her moment of inertia is MR^2. Since angular momentum is conserved the wheel must have equal and opposite angular momentum to her.
 
  • #3
Thanks so much! I was able to use that to get the right answer. Here's my final work:

moment of inertia (woman) = mr^2 = 240 kgm^2
angular speed of woman = v/r = 0.750 rad/s

Comservation of Angular momentum:
I(woman)(omega(woman)) = I(turntable)(omega(turntable))

So, Omega (turntable) = (I(woman)omega(woman)) / I(turntable)

= 0.360 rad/s

Thanks!
 

1. What is the conservation of angular momentum of a turntable?

The conservation of angular momentum of a turntable is a physical principle that states that the total angular momentum of a system remains constant unless acted upon by an external torque. In simpler terms, when a turntable is spinning, the angular momentum of the turntable and the record on it will remain the same unless an outside force is applied.

2. How is angular momentum conserved in a turntable?

Angular momentum is conserved in a turntable because the turntable and record have an initial angular momentum when the record is first placed on the turntable. As the turntable and record spin, any changes in their angular momentum must be balanced out by an equal and opposite change in the angular momentum of the rest of the system, such as the motor or the air around the turntable.

3. Why is the conservation of angular momentum important in turntables?

The conservation of angular momentum is important in turntables because it allows for a stable and consistent rotation of the record. Without this principle, the record would not be able to spin at a constant speed and could become unstable or even stop spinning altogether.

4. How does the conservation of angular momentum affect the sound quality of a turntable?

The conservation of angular momentum does not directly affect the sound quality of a turntable. However, it does play a role in maintaining a consistent rotation of the record, which can indirectly impact the sound quality. Any fluctuations in the rotation speed of the record can result in changes in pitch or sound quality.

5. Can the conservation of angular momentum be violated in turntables?

No, the conservation of angular momentum is a fundamental law of physics and cannot be violated. In turntables, the principle is always upheld as long as there are no external torques acting upon the system. Any changes in the rotation speed of the record can be attributed to external factors such as friction or motor speed, not a violation of the conservation of angular momentum.

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