What is the Lagrangian for a Mouse on a Rotating Wheel?

In summary, the conversation is about writing down the lagrangian for the motion of a mouse of mass m running around the inner circumference of a vertical wheel with mass M and moment of inertia I. The kinetic energy of the mouse is 1/2*m*a^2*θ^2 and the potential energy is V=mgacosθ. There is confusion about the rotational energy of the wheel, which is 1/2 Iω2.
  • #1
starryskiesx
6
0

Homework Statement


A mouse of mass m runs around the inner circumference of a vertical wheel which is free to rotate about the centre. The wheel has mass M and moment of inertia I. Let θ be the angle that the radius vector makes to the mouse from the downward vertical at time t. Write down the lagrangian for the motion.

Homework Equations


L = T - V

The Attempt at a Solution


So I can write down the kinetic energy of the mouse quite easily, in polar coordinates it's at constant ρ so T_mouse = 1/2*m*a^2*θ^2
Also the potential energy is V=mgacosθ.
The bit I'm confused about is the rotational energy of the wheel

Thanks!
 
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  • #2
hi starryskiesx! :smile:

(try using the X2 button just above the Reply box :wink:)
starryskiesx said:
1/2*m*a^2*θ^2

you mean 1/2*m*a2*ω2 :wink:
The bit I'm confused about is the rotational energy of the wheel

1/2 Iω2 :smile:
 

FAQ: What is the Lagrangian for a Mouse on a Rotating Wheel?

What is the Lagrangian Mouse on Wheel?

The Lagrangian Mouse on Wheel is a simple mechanical system used in physics and engineering to illustrate principles of classical mechanics, such as conservation of energy and Lagrange's equations of motion.

How does the Lagrangian Mouse on Wheel work?

The system consists of a mouse (represented by a point mass) on a horizontal wheel, which is free to rotate about its axis. The mouse is connected to the wheel by a spring, which acts as a restoring force. The motion of the mouse and the wheel can be described using Lagrange's equations, which take into account the kinetic and potential energies of the system.

What are the applications of the Lagrangian Mouse on Wheel?

The Lagrangian Mouse on Wheel can be used to study the oscillations and energy transfer between two connected bodies, as well as the effects of changing parameters such as mass, spring stiffness, and initial conditions. It is also a useful tool for understanding more complex mechanical systems.

How is the Lagrangian Mouse on Wheel different from other systems?

The Lagrangian Mouse on Wheel is a simplified version of more complex systems, making it easier to understand and analyze using mathematical principles. It is also a good representation of many real-world systems, such as a mass on a spring or a pendulum, which can be described using the same equations.

What are the limitations of the Lagrangian Mouse on Wheel?

While the Lagrangian Mouse on Wheel is a useful tool for understanding basic principles of classical mechanics, it is a highly idealized system and does not take into account factors such as friction, air resistance, and other external forces. These factors can significantly affect the behavior of real-world systems and must be considered in more complex models.

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