What Is the Force of Constraint in a Simple Pendulum?

[w3390]

Homework Statement

A simple pendulum has a mass M attached at the end of a massless rod of length L. Find the force of constraint the rod exerts on the bob.

Homework Equations

The Attempt at a Solution

It seems easy enough that the mass is constrained by the tension the rod exerts on the mass. Therefore, T = mgcos(theta). However, isn't the equation of constraint supposed to help you eliminate a variable when going through the Lagrangian to find the equation of motion?

 

[hikaru1221]

Therefore, T = mgcos(theta).

Don't jump into conclusion too fast. This is only true when the pendulum is at rest. When it swings, it gains centripetal acceleration...

 

[w3390]

Okay, so when swinging:

T - Mgcos(theta) = Mv^2/L

I still don't see how this equation helps at all.

 

[hikaru1221]

v can be found by using energy conservation law, right?
Even if Lagrangian method is applied, the result is the same.

 

[w3390]

You mean like 1/2Mv^2 = Mg(L - Lcos(theta))

V = sqrt[2g(L - Lcos(theta))]

 

[hikaru1221]

Mostly like that. The exact one should be: mv^2/2 - mgLcos(theta) = E, where E is the total (initial) mechanical energy of the pendulum.

 

[w3390]

So I still don't see what the point of finding this equation was. I was able to find the Lagrangian and go through and solve for the equation of motion all without having this equation.

 

[hikaru1221]

What I referred to is Newtonian method, based on force and energy analysis. Lagrangian method will still yield the same result, but you will still have to go through force analysis process since Lagrangian method avoids the forces of constraint (so here, Lagrangian method will help you find speed v, equivalent to the conservation energy equation). However, regardless of the method you use, the answer T=mgcos(theta) is wrong.