Conceptual question on angular momentum and Emech.

In summary, the question is whether the mechanical energy of a system consisting of a woman walking on a turntable is conserved. The solution manual argues that it is not, as the internal forces between the woman and the turntable do positive work, converting chemical energy to kinetic energy. As for the question of momentum conservation, the woman-turntable system is not conserved due to the fixed axis of the turntable, and the force exerted by the woman causing a change in the momentum of the system.
  • #1
RoboNerd
410
11

Homework Statement


A 60.0 kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 500 and radius 0f 2.00 m.

Turntable is initially at rest and is free to rotate around frictionless vertical axle through its center. Woman then starts walking around the rim at the constant speed of 1.50 m/s relative to the earth.

OK.

Is mechanical energy of the system conserved?

Homework Equations


Conservation of mechanical energy

The Attempt at a Solution


[/B]
OK. So my woman here is exerting a force pushing on the disk and the disk is pushing a force on the woman. These are equal and opposite forces... and are internal to the system.

I said that mechanical energy is conserved as there are only forces internal to the system.

My solutions manual says the following, however:
"The mechanical energy of this system is not conserved because the internal forces, of the woman pushing backward on the turntable and of the turntable pushing forwards on the woman, both do positive work, converting chemical to kinetic energy."

Huh? The two forces are internal... and thus do not affect the mechanical energy... or do they?

Who is right: the solutions manual, or I? If the solutions manual is right... then why is their approach right with the term "positive work?"

Thanks in advance.
 
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  • #2
Ohh.. and another question:

Is momentum conserved? I said yes. My solution manual disagrees and says the following:

"Momentum of the woman-turntable system is not conserved. The turntable's center of mass is always fixed. The turntable always has zero momentum [BECAUSE IT IS NOT MOVING TRANSLATIONALLY?]. The woman starts walking north, gaining northward momentum. Where does it come from? She pushes south on the turntable. Its axle holds it still against linear motion by pushing north on it, and this outside force delivers northwards linear momentum into the system."

Huh? This is a bit confusing... could anyone explain in simpler terms why or why not the momentum is conserved?

Thanks!
 
  • #3
RoboNerd said:

Homework Statement


A 60.0 kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 500 and radius 0f 2.00 m.

Turntable is initially at rest and is free to rotate around frictionless vertical axle through its center. Woman then starts walking around the rim at the constant speed of 1.50 m/s relative to the earth.

OK.

Is mechanical energy of the system conserved?

Homework Equations


Conservation of mechanical energy

The Attempt at a Solution


[/B]
OK. So my woman here is exerting a force pushing on the disk and the disk is pushing a force on the woman. These are equal and opposite forces... and are internal to the system.

I said that mechanical energy is conserved as there are only forces internal to the system.

My solutions manual says the following, however:
"The mechanical energy of this system is not conserved because the internal forces, of the woman pushing backward on the turntable and of the turntable pushing forwards on the woman, both do positive work, converting chemical to kinetic energy."

Huh? The two forces are internal... and thus do not affect the mechanical energy... or do they?

Who is right: the solutions manual, or I? If the solutions manual is right... then why is their approach right with the term "positive work?"

Thanks in advance.

Not exactly sure with your second question involving the momentum not being conserved, but in regards to the first question:
Mechanical energy is the sum of kinetic and potential energies. So think, what is the initial kinetic and potential energies compared to the final kinetic and potential energies?
 
  • #4
Conservation of momentum is the left null space of the linear system?
 
  • #5
Edited.
 
  • #6
RoboNerd said:
Huh? This is a bit confusing... could anyone explain in simpler terms why or why not the momentum is conserved?
You understand that we are discussing linear momentum here, not angular momentum, right?
As the explanation says, the turntable cannot acquire any linear momentum since its axis is fixed. Yet the woman exerts a force on it. The explanation has to be an equal and opposite force from somewhere (but perhaps with a different line of action, i.e. parallel).
As the text says, that equal and opposite force must come from the axle. If the woman accelerates to the north, the axle must supply a force to the north to stop the turntable's centre heading south.

For the first question, two parts of a system may exert equal and opposite forces on each other. Since that sum is zero, momentum is conserved: ##\int F.dt +\int -F.dt =0##. But for work the equation becomes ##\int F.ds+\int -F.(-ds)=2\int F.ds##.
As mattbeatlefreak noted, that energy can come from chemical energy, which is not counted as potential energy.
 
  • #7
mattbeatlefreak said:
Not exactly sure with your second question involving the momentum not being conserved, but in regards to the first question:
Mechanical energy is the sum of kinetic and potential energies. So think, what is the initial kinetic and potential energies compared to the final kinetic and potential energies?
Initially... we have zero. Afterwards, we have kinetic energy of rotation... mechanical energy is not conserved
 
  • #8
I see now. Thanks a lot everyone!
 

Related to Conceptual question on angular momentum and Emech.

1. What is angular momentum?

Angular momentum is a measure of the rotational motion of an object. It is defined as the product of an object's moment of inertia and its angular velocity.

2. How is angular momentum related to energy?

Angular momentum is conserved, meaning it remains constant unless an external torque is applied. This also means that it is directly related to energy, as changes in angular momentum correspond to changes in kinetic energy.

3. What is the difference between angular momentum and linear momentum?

Angular momentum is a measure of rotational motion, while linear momentum is a measure of straight-line motion. Linear momentum is the product of an object's mass and its velocity, while angular momentum is the product of its moment of inertia and angular velocity.

4. How is angular momentum related to Emech?

Emech, or mechanical energy, is the sum of an object's kinetic and potential energies. Angular momentum is related to Emech through the concept of conservation of energy, as changes in angular momentum can result in changes in Emech.

5. Can angular momentum be negative?

Yes, angular momentum can be negative. This is because it is a vector quantity and can have both magnitude and direction. A negative angular momentum indicates that the object is rotating in the opposite direction of its initial motion.

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