Equation of motion for a pendulum using Lagrange method

[morpheus343]

Homework Statement

Find the equation of motion of the pendulum

Relevant Equations

Lagrangian L=T-V
Kinetic-Potential energy

I move the system by a small angle θ . I am not sure if my calculations of kinetic and potential energy are correct.

 

[Orodruin]

What calculations? All you have given is the final result, which does not look correct to me.

 

[morpheus343]

I do not understand if this just a problem treated like a pendulum with a simple sphere but just adding the other 2 sphere's masses. Is the potential energy supposed to have a different y for each sphere?

 

[Orodruin]

You are asserting that the speed of the outer spheres are the same as that of B? Is this reasonable?

Is the potential energy supposed to have a different y for each sphere?

What would you say based on your second diagram in the OP? Are all spheres at the same height? Does the answer matter?

 

[morpheus343]

Then it would be like this. This was my first assumption but i got lost in the algebra and thought i was maybe wrong and it was supposed to be simpler

 

[Orodruin]

Well, for the potential energy, perhaps there is an argument to make it simpler? What can you say about the center of mass?

For the kinetic energy, I did not check the math, but yes, you indeed need to take the distance to C and D using the Pythagorean theorem. Note that, since the distance appears squared in T, you should probably keep the exact rational 1+4/9 = 13/9 in the expression.

 

[morpheus343]

Well, for the potential energy, perhaps there is an argument to make it simpler? What can you say about the center of mass?

For the kinetic energy, I did not check the math, but yes, you indeed need to take the distance to C and D using the Pythagorean theorem. Note that, since the distance appears squared in T, you should probably keep the exact rational 1+4/9 = 13/9 in the expression.

It's at sphere B which is the midpoint of CD. So i only need the vertical distance from the centre of mass when calculating the potential energy?

 

[Orodruin]

It's at sphere B which is the midpoint of CD. So i only need the vertical distance from the centre of mass when calculating the potential energy?

Yes. By definition, the potential energy for any mass distribution is given by
$$
V = \int U(z) \rho(\vec x) dV
$$
where $U(z)$ is the gravitational potential and $\rho$ the density. Assuming $U(z) = gz$ (homogeneous field), you’ll find
$$
V = g \int z \rho(\vec x) dV \equiv m g \bar z
$$
where $\bar z$ is the z-coordinate of the center of mass by definition:
$$
\bar z = \frac 1m \int z \rho(\vec x) dV
$$