Pendulum on a spring accelerating upward

In summary, the pendulum on Earth consists of a mass m suspended on a massless spring with equilibrium length d and spring constant k. The Hamiltonian of the system can be derived using T + U, or by finding the momenta and using them in the Hamiltonian equation. Energy is conserved in this system, as the equations of motion are derived from the Lagrangian which takes into account the total energy. The frequencies of small oscillations can be solved for by finding the Lagrangian equations of motion and taking the first order in the variables.
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Homework Statement



A pendulum on Earth consists of a mass m suspended on a massless spring with equilibrium length d and spring constant k. The pendulum point of support is moving up with constant acceleration a.

What is the Hamiltonian of the system? Derive the Hamiltonian equations of motion for the system. Discuss the relationship between the Hamiltonian and the total energy of the system (is energy conserved?). What are the frequencies of small (first order in the variables) oscillations for the system?


Homework Equations



L = T - U
H = T + U


The Attempt at a Solution



I'm using r, θ, and z as my coordinates.
T = 1/2 m [ (rθ'cosθ + r'sinθ)^2 + (-rθ'sinθ + r'cosθ + z')
U = mg (z + rcosθ)

L = T - U

The first part of the problem asks for the Lagrangian equations of motion. I found those by applying Euler's equation to L and solving the differential equations.

Now what is the correct method for finding H? Is it as simple as T + U? Or do I need to derive the momenta for the system (and how would I do that?)
 
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? And if I do find the momenta, how would I use them in my Hamiltonian?For the second part of the problem, I'm guessing that energy is conserved since the equations of motion are derived from the Lagrangian which takes into account the total energy. The third part of the problem is asking for the frequencies of small (first order in the variables) oscillations. I'm not sure how to solve this. Any help would be appreciated. Thanks!
 

1. What is a pendulum on a spring accelerating upward?

A pendulum on a spring accelerating upward refers to a physical system in which a mass attached to a spring is suspended from a fixed point and is subjected to an upward acceleration. This type of system is commonly studied in physics and can be observed in various real-world scenarios, such as amusement park rides.

2. How does the acceleration affect the motion of the pendulum?

The acceleration affects the motion of the pendulum by changing the equilibrium position of the system. As the system experiences an upward acceleration, the equilibrium point of the pendulum will shift upwards, causing the pendulum to oscillate at a higher point than it would in a stationary system.

3. What factors influence the period of a pendulum on a spring accelerating upward?

The period of a pendulum on a spring accelerating upward is influenced by the mass of the pendulum, the spring constant of the spring, and the acceleration of the system. As these factors change, the period of the pendulum will also change.

4. How is the period of a pendulum on a spring accelerating upward calculated?

The period of a pendulum on a spring accelerating upward can be calculated using the formula T=2π√(m/k+g/a), where T is the period, m is the mass, k is the spring constant, g is the acceleration due to gravity, and a is the upward acceleration of the system.

5. What is the relationship between the acceleration and the frequency of the pendulum?

The relationship between the acceleration and the frequency of the pendulum is that as the acceleration increases, the frequency of the pendulum will also increase. This means that the pendulum will oscillate more times per second as the acceleration of the system increases.

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