Conservation of angular momentum and change in rotational kinetic energy

In summary: The rotational kinetic energy increase by a factor of 4 also. This is because the rotational kinetic energy is computed by multiplying the mass's rotational velocity by the mass's moment of inertia. In this case, because the rotational velocity has been quadrupled, the rotational kinetic energy has also been quadrupled.
  • #1
chamddol
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I have a question regarding angular momentum and rot kinetic energy. For example, if angular momentum is conserved, and the radius is cut in half, then moment of inertia is reduced by a fourth, which will result in increase in angular velocity by factor of 4. My question is why is the rotational kinetic energy increase by a factor of 4 also. Since the equation of rot kinetic energy is (1/2)Iw^2, the new w increased by factor of 4 but the new I also decreased by factor of 1/4. So wouldn't KE stay the same?
 
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  • #2
welcome to pf!

hi chamddol! welcome to pf! :smile:
chamddol said:
… if angular momentum is conserved, and the radius is cut in half, then moment of inertia is reduced by a fourth, … why is the rotational kinetic energy increase by a factor of 4 also … wouldn't KE stay the same?

as you know, angular momentum is always conserved (if there's no external torque, of course)

in a "collision" situation, (mechanical) energy usually isn't conserved, but if the changes are gradual (as here), yes we would usually expect it to be conserved

however, that's forgetting the work energy theorem … work done = change in mechanical energy

imagine that you're rotating on ice, and you're holding onto a heavy mass on a rope

if you pull the rope in, the total energy increases because you are doing work (force "dot" distance) by pulling the mass in :wink:

when you reduce the moment of inertia of any rotating mass, you have to do work! :smile:
 
  • #3
chamddol said:
Since the equation of rot kinetic energy is (1/2)Iw^2, the new w increased by factor of 4 but the new I also decreased by factor of 1/4. So wouldn't KE stay the same?

Er - no. Think that through again. You've reduced I by a factor of 4, and increased w by a factor of 4, so you've increased w^2 by a factor of ...?
 
  • #4
chamddol said:
I have a question regarding angular momentum and rot kinetic energy. [...] radius is cut in half, [...] will result in increase in angular velocity by factor of 4. My question is why is the rotational kinetic energy increase by a factor of 4 also.

attachment.php?attachmentid=46972&stc=1&d=1336163618.png


Expanding on the answer given by tiny-tim:
The curved line in the diagram represents the trajectory of an object that is pulled closer to the center. The dark grey arrow represents the centripetal force.

Now, in the case of perfectly circular motion the centripetal force is at all times perpendicular to the instantaneous velocity, and hence there is no change of kinetic energy. But here, with the object being pulled closer to the center, the exerted force is not perpendicular to the instantaneous velocity.

You can think of the force as decomposed, one component perpendicular to the instantaneous velocity and one paralllel to the instantaneous velocity. The perpendicular component causes change of direction, the parallel component causes acceleration.

So in the process of pulling closer to the center you are doing work upon the object. This is big: by the time you've managed to reduce the radial distance by half you have quadrupled the kinetic energy.


In general: when you have an object in circumnavigating motion, and you pull that object closer to the center, then the acceleration occurs because you are doing work upon that object.
 

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  • #5
haruspex is right. Its a simple mathematical error.
 

What is conservation of angular momentum?

Conservation of angular momentum is a fundamental law of physics that states that the total angular momentum of a system remains constant, unless acted upon by an external torque. This means that the angular momentum of a system cannot be created or destroyed, only transferred or transformed.

How does an object's angular velocity affect its angular momentum?

An object's angular velocity, which is the rate at which it rotates, plays a crucial role in determining its angular momentum. The greater the angular velocity, the greater the angular momentum, assuming the object's mass and distribution of mass remain constant.

What is the relationship between angular momentum and rotational kinetic energy?

Angular momentum and rotational kinetic energy are closely related, as they both involve the rotation of an object. The angular momentum of an object can be thought of as its "rotational momentum", while rotational kinetic energy is the energy associated with an object's rotation. As an object's angular momentum changes, so does its rotational kinetic energy.

Can the total angular momentum of a system change over time?

No, according to the law of conservation of angular momentum, the total angular momentum of a system remains constant over time. This means that if one part of the system experiences a change in angular momentum, another part must experience an equal and opposite change in order to maintain the total angular momentum of the system.

How does an external torque affect the conservation of angular momentum?

An external torque, which is a force that causes an object to rotate, can affect the conservation of angular momentum. If an external torque acts on a system, it can change the total angular momentum of the system by transferring angular momentum from one part to another. However, the total angular momentum of the system will remain constant, as long as no other external torques are present.

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