Looking for a unified expression for energy in rotating systems

  • #1
GeneralJez
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TL;DR Summary
I am looking for a unified expression for energy in rotating systems
Hello,
I am wondering if anyone can give me a hand.
I am looking for a unified expression for energy in rotating systems or wondering if one even exists.
Any help or equations you wish to share about the topic would be great.
Sorry I am quite new to physics.

Thanks
 
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  • #2
GeneralJez said:
TL;DR Summary: I am looking for a unified expression for energy in rotating systems

Hello,
I am wondering if anyone can give me a hand.
I am looking for a unified expression for energy in rotating systems or wondering if one even exists.
Any help or equations you wish to share about the topic would be great.
Sorry I am quite new to physics.

Thanks
Welcome to PF. This should get you started:

https://en.wikipedia.org/wiki/Rotational_energy
 

1. What is a unified expression for energy in rotating systems?

A unified expression for energy in rotating systems typically involves the sum of translational kinetic energy, rotational kinetic energy, and potential energy if applicable. The general formula can be expressed as E_total = K_trans + K_rot + U, where K_trans = 1/2 mv² (translational kinetic energy), K_rot = 1/2 Iω² (rotational kinetic energy, with I being the moment of inertia and ω the angular velocity), and U represents the potential energy.

2. How is the moment of inertia (I) calculated in these systems?

The moment of inertia, I, depends on the mass distribution relative to the axis of rotation. It is calculated as I = ∫r² dm, where r is the distance of a mass element dm from the axis of rotation. For discrete mass distributions, it can be calculated as I = Σm_ir_i², where m_i is the mass and r_i is the distance of each mass element from the axis of rotation.

3. What role does angular velocity (ω) play in the energy of rotating systems?

Angular velocity, ω, is crucial in determining the rotational kinetic energy of a system. It represents the rate of rotation around an axis, and the rotational kinetic energy is directly proportional to the square of angular velocity. Thus, as angular velocity increases, the rotational kinetic energy increases quadratically, making it a key factor in the overall energy of rotating systems.

4. Can potential energy be ignored in rotating systems?

Potential energy in rotating systems should not be ignored if there are forces acting through a distance, such as gravitational, elastic, or other field forces. For instance, in a pendulum that rotates around a pivot, gravitational potential energy plays a significant role in the system's total energy. However, in scenarios strictly dealing with rotational motion in a constant gravitational field or no field, potential energy might be less relevant or constant.

5. How does conservation of energy apply to rotating systems?

In an isolated rotating system, the law of conservation of energy states that the total energy remains constant if no external torques or forces do work on the system. This means that the sum of kinetic (both translational and rotational) and potential energies must remain constant throughout the motion, allowing for energy transformations between these forms but not for any increase or decrease in total energy.

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