Calculating Time for Object to Hit the Floor in a Pulley System

[Icetray]

Hi guys,

I have this question below but I don't have the answer to it and I'm not sure if what I am doing is right. I attempted the solution by balancing the forces as I am not as clear on how to use the energy method yet.

Is my solution/answer correct? (:

Thanks guys!Homework Statement

Consider the system shown in Figure 10.42 with m1 = 20.0kg, m2 = 12.5kg, R(radius of pully)=0.2m and mass of uniform pulley M=5.00kg. Object m2 is resting on the floor and m1 is 4.00 meters above the floor when it is released from rest. The pully axis is frictionless. Calculate the time for m1 to hit the floor.

Relevant equations and attempt at solution
M1:
T$_{1}$ = m$_{1}$g - m$_{1}$a
T$_{1}$ = 196.2 - 20a

M2:
T$_{2}$ = m$_{2}$g + m$_{2}$a
T$_{2}$ = 122.625 + 12.5a

Pully:
T$_{1}$ - T$_{2}$ = I$\alpha$
T$_{1}$ - T$_{2}$ = (0.5)(MR$^{2}$)(a/R)
T$_{1}$ - T$_{2}$ = 0.5(5.00)(2.00)a

Subbing in T$_{1}$ and T$_{2}$ I get a = 2.23m/s$^{2}$

Then I use Xf = Xi + Ut +0.5at$^{2}$ to get t and get 1.89s

Is this right?

 

[gneill]

Remember that torque is F x L, where F is the applied force and L is the "lever arm" or radius vector to the center of rotation. So check your Pulley section to make sure that you're equating torques with torques.

 

[Icetray]

Remember that torque is F x L, where F is the applied force and L is the "lever arm" or radius vector to the center of rotation. So check your Pulley section to make sure that you're equating torques with torques.

Thanks for your reply! (:

So:
T$_{1}$ - T$_{2}$ = I$\alpha$

Should be:
T$_{1}$R - T$_{2}$R = I$\alpha$

Right? (:

 

[gneill]

Thanks for your reply! (:

So:
T$_{1}$ - T$_{2}$ = I$\alpha$

Should be:
T$_{1}$R - T$_{2}$R = I$\alpha$

Right? (:

Yup.

 

[Icetray]

Yup.

Just one more question. If the cylinder was a hollow one I understand that:
I = (0.5)M(R$_{1}$$^{2}$ - R$_{2}$$^{2}$)

Does my $\alpha$ = a/(R$_{1}$ - R$_{2}$) as well?

What about my $\tau$? Does it become:
(R$_{1}$-R$_{2}$)(T$_{1}$- T$_{2}$) as well?

Thanks! (:

 

[gneill]

You don't need to change anything but the moment of inertia unless the radius at which the torque is applied changes too.

The moment of inertia is the rotational analog of mass. In a linear system, if you had a hollow sphere (spherical shell) rather than a solid one you'd have to do something similar in order to calculate the mass of the object, but after that you'd just use the calculated mass and get on with things.

 

[Icetray]

You don't need to change anything but the moment of inertia unless the radius at which the torque is applied changes too.

The moment of inertia is the rotational analog of mass. In a linear system, if you had a hollow sphere (spherical shell) rather than a solid one you'd have to do something similar in order to calculate the mass of the object, but after that you'd just use the calculated mass and get on with things.

So the radius I use for the calculation of $\tau$ and $\alpha$ will be that of the full radius (i.e. from att the way out to the centre) and the only thing is actually affects is the calculation of the inertia of the disk? :D

 

[gneill]

So the radius I use for the calculation of $\tau$ and $\alpha$ will be that of the full radius (i.e. from att the way out to the centre) and the only thing is actually affects is the calculation of the inertia of the disk? :D

Yes. The torque calculation always involves the actual distance from the center of rotation to where the force is applied.

 

[Icetray]

Yes. The torque calculation always involves the actual distance from the center of rotation to where the force is applied.

Thanks! You're a LIFESAVER! (: