Help with rotation of rigid bodies

[RussG]

Alright, I'm stuck on this problem.

Two metal disks, Radius 1 = 2.5cm, Mass 1 = 0.8kg and Radius 2 = 5.0cm, Mass 2 = 1.6kg are wielded together and mounted on a frictionless axis through their common center.

The first part asked what is the total moment of inertia of the two discs, which I got and it was 2.25 x 10^-3 kgm^2

Part b attaches a string to the smaller disk holding a cylinder of mass 1.5kg 2m above the ground. It asks the velocity of the cylinder right before it hits the ground when it's released from rest.

Then it moves the string in part c to the larger disc. I have the solutions but am not getting the same answers when I attempt to solve for it. I know that when attached to the larger disc it will have a greater speed because of the greater mass since their radiuses do not matter.

I have tried using the good ol K1 + U1 = K2 + U2 to determine its velocity.

I know K1 at rest is 0, and U1 is mgh. I know U2 is 0. So I need the final kinetic energy, right? Well the final kinetic energy would be 1/2mv^2 + 1/2Iw^2, or am I getting something wrong here?

since w = v/R I substituted that in.

I get 1/2(1/2MR^2)(v/R)^2 + 1/2mv^2 = mgh

solving for v I get sqrt((2gh)/(1+M/2m)) = v

Plugging in the numbers I'm not coming up with the correct velocities. Where have I gone wrong? Thanks for any help!

 

[member 553083]

You have almost got it correct! You need to take into account the mass of the disks when calculating the final kinetic energy. The equation for final kinetic energy should be: K2 = 1/2(M1 + M2 + MR^2)(v/R)^2 + 1/2mv^2 Where M1 and M2 are the masses of the two disks and MR^2 is the moment of inertia of the two disks. Once you have this equation, you can proceed as you did before to find the velocity of the cylinder right before it hits the ground.

 

[wais]

Hi there,

It seems like you are on the right track, but there may be a few errors in your calculations. First, when using the equation K1 + U1 = K2 + U2, make sure that you are consistent with your units. In this case, since the masses were given in kilograms, make sure to use joules for energy and meters for height.

Next, when calculating the final kinetic energy, you should use the moment of inertia of the entire system (both disks and the cylinder attached) since they will all be rotating together. So the equation would be 1/2(I1 + I2 + Icyl)v^2, where Icyl is the moment of inertia of the cylinder about the axis of rotation.

Also, when substituting w = v/R, make sure to use the radius of the entire system, which would be the sum of the radii of the two disks.

Lastly, when plugging in numbers, make sure to convert all units to their base units (e.g. cm to m) to get consistent results.

I hope this helps and good luck with your problem!