Finding the tension of a rope given the mass of a pulley

[Jacobs]

Homework Statement

mass of pulley is 8 kg

Homework Equations

m1a=T
m2g-T=m2a

The Attempt at a Solution

I solved question with neglecting mass of pulley but should I?

 

[Chandra Prayaga]

How does the mass of the pulley affect the problem?

 

[Dr Dr news]

If you want to include the rotational inertia of the pulley, you either need to know its radius or solve for the acceleration in terms of the radius.

 

[Jacobs]

Is that right?

 

[haruspex]

Is that right?

That is correct.

 

[Jacobs]

But why 1/2mr^2 instead of mr^2?Yes it is cylinder but particle moves as x and y-axis not z

 

[haruspex]

But why 1/2mr^2 instead of mr^2?Yes it is cylinder but particle moves as x and y-axis not z

Not all parts of the pulley are at distance r from the axis.

 

[Jacobs]

But r is radius of pulley

 

[Dr Dr news]

To find the inertia of the pulley or disk, you need to integrate r^2 dm over the dimensions of the pulley I = ∫ r^2 dm = ∫∫∫ r^2 (ρ r dr dθ dz). If you do that, you will find I = (mr^2)/2 where m = ρV = ρ π r^2 t, t = thickness of the disk.

 

[haruspex]

But r is radius of pulley

Yes, but only its periphery is at distance r from the axis. A mass element dm on the oeriphery does have moment of inertia dm r² about the axis. Other parts of it are closer to the axis so have a smaller moment of inertia. The average turns out to be the same as if all parts were r/√2 from the axis. Dr dr news has posted the details of that.