Working out energy change during change of moment of inertia

[govinda]

theres a standard question about the spinning dancer who pulls her hands inward reducing her moment of inertia and increasing her ang. velocity . it seems she had to do some work against the centrifugal force (from momentum and energy equations)
i thought of a simpler example to check if the amt of work that needs to be done agianst the force is indeed equal to the change in energy .considering a particle insted of the ballerina moving in a circle i worked out the work necassary using integral of force times distance moved . i substiuted values for ang. vel. since it isn't constant(from cons of ang momentum eq.) and it worked out to be a logarthmic expression . the change in energy from the first approach was different( no log term) . i have a feeling i have made some fundamental mistake . the calculations seem ok.,
govinda

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"if its you against the world bet on the world "

 

[Doc Al]

I don't know how you got a logarithmic expression.

Take the simpler case of a mass on a string, rotating on a frictionless table. Let the string pass through a hole in the center of the table, which is the axis of rotation. The string is pulled from below, drawing the mass closer to the center. Calculate the work done by direct integration of the string tension and compare to the the change in KE found from conservation of angular momentum. They should be equal.

 

[charng]

Hello Govinda,

Thank you for sharing your thoughts and calculations on the change in energy during a change in moment of inertia. First of all, I would like to commend you for your critical thinking and for trying to understand this concept through different examples.

In regards to your simpler example, it is important to note that the integral of force times distance moved only applies to a constant force. In the case of a particle moving in a circle, the force is not constant as it changes direction continuously. Therefore, the work done in this scenario cannot be calculated using the simple integral formula.

When we look at the spinning dancer example, we can see that the work done against the centrifugal force is indeed equal to the change in energy. This is because the dancer is constantly pulling her hands inward, applying a force against the centrifugal force, and thus doing work. This work is then converted into kinetic energy, resulting in an increase in angular velocity.

I would suggest revisiting your calculations and considering the changing force in your simpler example. Additionally, it might be helpful to look at the concept of work and energy in rotational motion in more detail, as it can be a bit more complex than in linear motion.

Overall, keep up the critical thinking and don't be discouraged if you encounter challenges in understanding a concept. It's all part of the learning process. Best of luck in your studies!