# Lagrange equation problem involving disk

#### pentazoid

1. The problem statement, all variables and given/known data
A uniform disk of mass M and radius a can roll along a rough horizontal rail. A particle of mass m is suspended from the center C of the disk by a light inextensible string b. The whole system moves in a vertical plane through a rail. Take as generalized coordinates x, the horizontal displacement of C, and theta , the angle between the string and the downward vertical. Obtain Lagrange's equation . Show that x is a cyclic coordinate and find the corresponding conserved momentum p_x. Is p_x the horizontal linear momentum of the system?

Given that theta remains small in the motion , find the period of the small oscillations particle.

2. Relevant equations

Ihave to “linearize” the θ equation, after eliminating derivitives
of x, to get an equation for θ that is eﬀectively simple harmonic motion.
In this case, linearizing means assuming θ and ˙
θ are small, approximat-
ing all of the trigonometric functions up to terms linear in θ, and throwing
out terms quadratic in θ and ˙
θ. Soln.: the period for small oscillations is

q
3M b
.

3. The attempt at a solution

x_1=q_1-F y_1=0 z_1=0
x_2=q_1+a*sin(q_2)-F,y_2=0,z_1=-a*cos(q_2)

Not sure what to do after that

#### Dr.D

What does it mean when it says, "The whole system moves in a vertical plane through a rail."? What is "through" about in this sentence?

You really did not show any relevant equations, and more importantly, you did not show an attempt at a solution. Yes, you wrote something under that heading, but it is awfully hard to understand what that means, and it looks more like some sort of final result, rather than a process for getting there.

To work this problem, you need to start with a good figure, identify the necessary coordinates, write the kinematic relatiions and the constraint relation(s), and then formulate your equations very carefully. The solution will be more than two lines long, I can assure you.

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