Why do the works of two objects in rotation differ when friction is present?

[V711]

Hi,

I would like to compute sum of work for 2 objects in rotation, look at image please. The grey disk has a rotationnal velocity of Y rd/s. Red disks turn at X rd/s in the same direction. Red disk don't turn around itself.

X > Y

At time 't' friction is on.

The Work for torque = Torque * angle of rotation ( W=T*Theta )

Theta = 1/2 * a * t² with 'a' the rotationnal acceleration

Second law of Newton : Torque = I * a , with I the inertia

Angle of rotation = 1/2 * T/I * t²

Work of torque = 1/2 * T²/I * t²

Torque to red disks is the same in value than the torque of grey disk.

Work of grey disk = 1/2 * T²/I1 * t² with I1 the inertia of grey disk
Work of red disks = -1/2 * T²/I2 * t² with I2 the inertia of red disks

It seems like that the work lost by red disks (X > Y) is not the same than work won by grey disk IF inertia I1 is different of I2. Friction produce temperature but it's possible to imagine something else where there is no friction and works will be different too. So, for me, works must be the same in value.

Maybe I can't think like that, so could you explain ? If I want to use these formulas how can I do ?

Thank you

 

[CWatters]

I was not able to follow your explanation.

With friction ON the red disks are effectively glued to the grey disk. You can calculate the energy stored in the system if you calculate the moment of inertia for the combination.

Energy = 0.5 (I₁+I₂) w²

With friction OFF you have two separate parts. You can calculate moment of inertia for each and the energy stored in each.

Energy = 0.5 I₁ w² + 0.5 1₂ w²

You get the same answer because moments of inertia add up (if they are on the same axis) and the angular velocity (w) is the same.

 

[V711]

I'm agree with your explanation but I would like to know when friction apply a torque T and not "glue" all the system. Each object has inertia and rotationnal velocity, at a time 't' friction is ON and a torque is applied from one object to another, the object that has greater velocity increase rotationnal velocity of other. With formulas from my first post I can't understand what's wrong. I use Newton law and standard formulas of work I think. Energy is conserved so all energy lost by one object must be give to another, but not with formulas. Sure, there is friction and energy is transformed to heating, but it's possible to have a torque with a complex mechanical system where friction is 0, like that all energy from one object must be go to other object, I would like to understand with this case. I hope it's clear like that, tell me.

 

[CWatters]

Energy is conserved so all energy lost by one object must be give to another...

I don't understand which object looses energy and which gains energy in your system?

Are you starting with the grey disk spinning and the red disks stationary (friction OFF)?

Then you turn friction ON?

 

[V711]

Red disk and grey disks are turning at w1 and w2 rd/s with w1 and w2 positives, we give energy for that 1/2*I1*w1² + 1/2*I2*w2². Imagine a mechanical system that can give torque from one disk to other without friction, think like a theoretical problem. If w2 > w1 one disk will slow down and other increase its rotationnal velocity, one disk loose energy other won. Now, if I use formulas at first message, the energy must be the same. But this give:

1/2*I1*w1²+1/2*T²/I1*t² + 1/2*I2*w2²-1/2*T²/I2*t²

The difference is :

1/2*T²/I1*t² - 1/2*T²/I2*t²

I1 can be different of I2, the result is not 0. What's wrong ?

 

[CWatters]

Not sure if this is the only problem but..

Theta = 1/2 * a * t²

Should that be..

Theta = wi*t + 0.5*a*t²

where wi is the initial angular velocity. Wi can be zero one parts but not both.