# Kinetic Energy of Double Compound Pendulum and Parallel Axis

• Chrono G. Xay
In summary, a double compound pendulum has two pivoting points, and needs to use the parallel axis theorem to find the kinetic energy.
Chrono G. Xay
Kinetic Energy of Double Compound Pendulum and Parallel Axis Theorem

Hello, there.

I have a project I'm working on where I need to be able to calculate the kinetic energy of what basically amounts a double compound pendulum. However, the pivot point of the second pendulum is not at the center of mass (ergo the necessity of using the parallel axis theorem). Furthermore, because of the way the system is set up, the angles θ1 and θ2 are more like deviations from 'rest' positions that are not parallel with the force of gravity. I tried reading up on the equation for a double compound pendulum and the parallel axis theorem online, but nowhere, I believe, have I found an article combining the two concepts in the way I have in mind, not even counting the angle condition I was wanting to set up. Would anyone be able to give me some pointers? I'm not afraid to do a little reading, if need be.

(Note: Please know that I have a good familiarity with algebra. Calculus, not so much but I know just enough of that to maybe get an idea of what's going on.)

Do you have a drawing of your system? Please post a figure.

It's just a system of someone holding a drumstick, where only their fingers and wrist are seen as being able to pivot. Θ should probably instead be the acute angle on the other side, Θ and Φ are both the rest angles of their particular joint no greater than 90°/0.5π or less than 0 each, and only... increase from those values, if pivoting using that part of the hand, then return either Θ or Φ, respectively.

While I would LOVE to be able to extend this out to calculate what it would be like if the elbow was also pivoting (or just the elbow and fingers, with the wrist 'locked' either at rest or another angle), I've seen the equation for motion of a double compound pendulum, and I get the feeling that the complexity might increase exponentially, and then much, MUCH more if, say, the fingers and wrist were locked but the elbow and SHOULDER were to pivot instead (assuming, of course, that they were to all rotate in the same plane)...

## 1. What is the formula for calculating the kinetic energy of a double compound pendulum?

The formula for calculating the kinetic energy of a double compound pendulum is KE = 1/2 (m1v1^2 + m2v2^2), where m1 and m2 are the masses of the two pendulum bobs, and v1 and v2 are the linear velocities of the bobs.

## 2. How does the parallel axis theorem apply to the kinetic energy of a double compound pendulum?

The parallel axis theorem states that the moment of inertia of a body about an axis is equal to the moment of inertia of the body about a parallel axis through its center of mass, plus the product of the mass of the body and the square of the distance between the two axes. In the case of a double compound pendulum, this means that the kinetic energy will be greater when the bobs are further from the center of mass, and vice versa.

## 3. What factors affect the kinetic energy of a double compound pendulum?

The kinetic energy of a double compound pendulum is affected by the masses of the two bobs, the length of the pendulum arms, and the initial velocity of the bobs. Additionally, the angle at which the pendulum is released can also affect the kinetic energy.

## 4. How does the kinetic energy of a double compound pendulum change over time?

As the pendulum swings, the kinetic energy will change due to the changing velocity and positions of the bobs. At the bottom of the swing, when the bobs are moving the fastest, the kinetic energy will be at its maximum. At the top of the swing, when the bobs are momentarily stationary, the kinetic energy will be at its minimum.

## 5. How is the kinetic energy of a double compound pendulum related to its potential energy?

According to the law of conservation of energy, the total energy of a system remains constant. In the case of a double compound pendulum, the potential energy at the top of the swing is converted into kinetic energy at the bottom of the swing. As the pendulum continues to swing, this energy is constantly being converted back and forth between potential and kinetic energy.

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