Inertia

[gunnar]

Hi. I have a problem.
Two metal disks with different radius and different weight are welded together. The smaller disk has a string wrapped around it with a block on the other end. I have to calculate the speed of the block after 2 meters. I'm trying to use this equation: v=sqrt(2gh/1+M/2m) to solve this. Where M is the total mass of the disks and block while m is the mass of the block. This is not working. Do you guys know a way around this, I guess I need some kind of a relation between the two disks.

[Galileo]

I don't understand the entire problem. How are the disks set in motion?
Are they allowed to roll on a table with the block hanging over a pulley at the end of the table? Or something else?

[Evil_Kyo]

If i've understood correctly the two disks act as a pulley with the mass at the end of the string. If it is the case you can work it as a typical solid rigid problem calculating the angular acceleration and relating it with the linear acceleration of the mass. Then it becomes a "semi-free" fall problem.

Or you can take the simpler approach of conservation of energy:

Supposing the pulley is fixed so its potential energy remains constant, the initial energy of the system is

mgh_1

and the final would be

\frac{1}{2}(I_1+I_2)\omega^2+\frac{1}{2}mv^2+mgh_2

I_1 and I_2 are the moments of inertia of the disks.

you can relate v and \omega by v=\omega r

where r is the radius of the disk where the string is wrapped.

I think you can work the rest. I think there's no major mistake, feel free to correct me.

[gunnar]

You understod the probmlem I think but I'm not quite following you Evil_kyo. I only need one kind of h since it's falling from 0 to 2 m. I have been trying to solve this with conservation of energy. I can't see from you equation what the tangential speed v should be, is the equation equal to zero or mgh?

With thanks,

Gunnar

[prasanna]

I only need one kind of h since it's falling from 0 to 2 m. I have been trying to solve this with conservation of energy.



The only kind of h that you need is nothing but Evil_Kyo's h_1 - h_2(taking origin at the pulley)
Since your h_1 = 0 and h_2 = -2,

Your only kind of h = \0-(-2) = 2

I can't see from you equation what the tangential speed v should be, is the equation equal to zero or mgh?

Tangential speed cannot be mgh!!!

[Evil_Kyo]

As prasanna has noted i haven't selected a origin for the potential energy.
h_1 and h_2 mean the initial and final position of the falling mass attached to the pulley. You don't know it but you know its difference h_1-h_2 that equals the length traveled by the falling mass 2 metres.

Conservation of energy states that the initial energy of the system equals the final energy, then

mgh_1=\frac{1}{2}(I_1+I_2)\omega^2+\frac{1}{2}mv^2 +mgh_2

rearranging

mg(h_1-h_2)=\frac{1}{2}(I_1+I_2)\omega^2+\frac{1}{2}mv^2

now we have the equation relating the angular speed \omega of the pulley and the linear speed v of the falling mass.

v=\omega r

Imagine the pulley rotates an angle \omega in a unit of time, then the lenght of string unrolled is equal to the lenght of arc subtended by this angle s=\omega r. If the pulley rolls \omega in a unit of time the mass falls \omega r in a unit of time, then \omega r is the speed v of the mass.

Using this equation we can simplify

mg(h_1-h_2)=\frac{1}{2}(I_1+I_2)(\frac{v}{r})^2+\frac{1}{ 2}mv^2


mg(h_1-h_2)=\frac{1}{2}(\frac{I_1+I_2}{r^2}+m)v^2

v^2=\frac{2mgr^2(h_1-h_2)}{I_1+I_2+mr^2}

It's finished if there is no major mistake, i hope this helps you.

[gunnar]

It works, of course. Thanks a lot.
I love this place.