How does a disk roll down an incline?

In summary, the conversation discusses the dynamics of a uniform disk rolling down an inclined ramp with an angle of inclination q. The Lagrangian and Hamiltonian are derived using the generalized coordinate of the angle f. The Hamiltonian is defined as the sum of the kinetic and potential energy. Hamilton's equations of motion are then written and solved for the given scenario. Questions are raised about the specific values used in the equations, such as the parameter a and the definition of the Hamiltonian.
  • #1
Gogsey
160
0
A uniform disk of mass m and radius R rolls without slipping down a ramp inclined at angle q to the horizontal.
Using the angle f through which it turns as a generalised coordinate, write the lagrangian, and then the Hamiltonian.
Write out and solve Hamilton’s equations of motion.

Ok, so I'm not really sure what to do here at all. Is ther 2 parts to the kinetic enerdy of the disk, one due to the disk spinning, and the other due to moving down the incline?
 
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  • #2
So I have my Lagrangian as:

L = (m(a phidot)^2)/2 + mga Phi sin(theta)

I did use a an example to start me off so I'm not sure what a is. Is a just the radius, which would be R for this example?

Also, whta is the definition of the Hamiltonian? Is it just the total energy T + U?

Lastly, what is Hamilton's equation of motion? Is it just H = pdot q - L?
 

Related to How does a disk roll down an incline?

What is the initial velocity of a disk rolling down an incline?

The initial velocity of a disk rolling down an incline is determined by the height of the incline and the angle of the incline. The higher and steeper the incline, the greater the initial velocity will be.

What is the acceleration of a disk rolling down an incline?

The acceleration of a disk rolling down an incline is determined by the force of gravity, the angle of the incline, and the mass and shape of the disk. The acceleration can be calculated using the formula a = gsinθ, where g is the acceleration due to gravity and θ is the angle of the incline.

How does the mass of the disk affect its rolling down an incline?

The mass of the disk affects its rolling down an incline by influencing its acceleration. The greater the mass of the disk, the greater the force of gravity acting on it, resulting in a higher acceleration. However, the shape of the disk can also affect its acceleration, so a heavier disk may not necessarily roll faster than a lighter one.

What factors can affect the speed of a disk rolling down an incline?

The speed of a disk rolling down an incline can be affected by several factors, including the height and angle of the incline, the mass and shape of the disk, and any external forces acting on the disk (such as friction). In general, a steeper incline and a lighter, more streamlined disk will result in a faster speed.

Can the speed of a disk rolling down an incline be calculated?

Yes, the speed of a disk rolling down an incline can be calculated using the formula v = √(2ghsinθ), where v is the speed, g is the acceleration due to gravity, h is the height of the incline, and θ is the angle of the incline. This formula assumes that there is no external force acting on the disk, such as friction.

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