# Solving Cylinder+Bead System Problem | Hello.

• LCSphysicist
In summary: Oops. Forgot a thing: The curve itself described probably has the shape ##r = r_{o}, f(z,\theta)=0##. So tecnically, we have just two coordinates if the cylinder is rotating? ##\theta, \phi##?Yes.
LCSphysicist
Homework Statement
A uniform cylinder of radius a and density ρ is mounted so as to rotate freely
about its axis, which is vertical. On the outside of the cylinder is a rigidly fixed uniform spiral
or helical track along which a point mass m can slide without friction. Suppose the point mass
starts at rest at the top of the cylinder and slides down under the influence of gravity.
FInd the lagrangian of the (cylinder+bead) system
Relevant Equations
.
Hello. I am having some difficulties regarding this problem.

I am not sure, actually, how to relate the angle rotated by the cylinder ##\phi## and the angle drawed by the bead itself ##\theta##.
I have been thinking in two alternatives:

a) The first one is that, since there are no friction forces, the angle are independent. So the Kinect energy of the whole system would be $$\frac{I \dot \phi ^2}{2} + \frac{m(\dot z ^2 + (r (\dot \theta + \dot \phi) ) ^2)}{2}$$

b) The second alternative would be to consider that, since gravity itself acts only at the z direction, and the rest of forces are internal. ##L_{z (cylinder)} = L_{z( bead)}##

I am not sure which one is correct, if neither.

Is this the full problem formulation? There seems to be no actual question involved.

Your LaTeX is also missing a brace.

LCSphysicist
Orodruin said:
Is this the full problem formulation? There seems to be no actual question involved.

Your LaTeX is also missing a brace.
I just forgot to add the question hahah. I should need the lagrangian. Since the kinect term is the main problem here, i am not even talking about lagrangian itself.

My Latex seems normal here, please att the page to see if it changes.

a) You need to introduce an appropriate number of generalised coordinates for your system. How many coordinates would be required in order to specify the system’s configuration?

b) I am unsure what you are trying to say here. It is unclear if L refers to some Lagrangian or to angular momentum about the z axis.

Herculi said:
My Latex seems normal here, please att the page to see if it changes.
For some reason it was displayed as a yellow box saying brace missing. It seems ok upon reload..

LCSphysicist
Orodruin said:
a) You need to introduce an appropriate number of generalised coordinates for your system. How many coordinates would be required in order to specify the system’s configuration?

b) I am unsure what you are trying to say here. It is unclear if L refers to some Lagrangian or to angular momentum about the z axis.For some reason it was displayed as a yellow box saying brace missing. It seems ok upon reload..
a) I think there are three independent coordinates, supposing that the cylinder and the bead rotates independently. So the coordinates would be:
Angle rotated by the cylinder ##\theta##
Angle rotated by the bead ##\phi##
z distance traced by the bead ##z##

b)If this condition is right, there would be just two coordiantes. L means angular momentum

The problem is that i am not sure which, if any, is right

Herculi said:
a) I think there are three independent coordinates, supposing that the cylinder and the bead rotates independently. So the coordinates would be:
Angle rotated by the cylinder θ
Angle rotated by the bead ϕ
z distance traced by the bead z
Think about the following: What if the cylinder was fixed, how many independent coordinates would you have? How many are added by allowing the cylinder to rotate?

Herculi said:
b)If this condition is right, there would be just two coordiantes. L means angular momentum
Angular momentum conservation comes in as a first integral of the equations of motion. That comes a couple of steps later unless you are looking for effective descriptions which you do not seem to be.

LCSphysicist
Orodruin said:
Think about the following: What if the cylinder was fixed, how many independent coordinates would you have? How many are added by allowing the cylinder to rotate?Angular momentum conservation comes in as a first integral of the equations of motion. That comes a couple of steps later unless you are looking for effective descriptions which you do not seem to be.
Oops. Forgot a thing: The curve itself described probably has the shape ##r = r_{o}, f(z,\theta)=0##. So tecnically, we have just two coordinates if the cylinder is rotating? ##\theta, \phi##?
I am sorry, i am always confused when i come across problems involving two angles. I am not sure when the angles are independent or not. But since here there are no friction, i guess they are independent?
So is a) right?, if we replace ##\dot z## by ##\frac{dz}{d \theta} \dot \theta##?

Yes.
Also: Since the spiral is uniform, z is a linear function of ##\phi##.

LCSphysicist

## What is the Cylinder+Bead System Problem?

The Cylinder+Bead System Problem is a physics problem that involves a cylinder with a bead placed on top of it. The goal is to determine the motion of the bead as the cylinder rotates.

## What are the key concepts involved in solving the Cylinder+Bead System Problem?

The key concepts involved in solving this problem are rotational motion, centripetal force, and friction. These concepts are used to analyze the forces acting on the bead and determine its motion.

## What are the steps to solve the Cylinder+Bead System Problem?

The steps to solve this problem are: 1) Identify the forces acting on the bead, including gravity, normal force, and friction. 2) Use Newton's second law to write equations of motion for the bead in the x and y directions. 3) Apply the conditions for rotational equilibrium to determine the net torque on the bead. 4) Use the equations of motion and torque to solve for the acceleration and angular acceleration of the bead. 5) Use these values to determine the motion of the bead.

## What are some common mistakes when solving the Cylinder+Bead System Problem?

Some common mistakes when solving this problem include forgetting to include all the forces acting on the bead, not considering the direction of the forces, and not properly setting up the equations of motion and torque.

## How can I check my solution to the Cylinder+Bead System Problem?

You can check your solution by analyzing the physical reasonableness of the results, such as whether the bead's motion matches the expected behavior. You can also compare your solution to known solutions or use computer simulations to verify your results.

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