How Do You Calculate Angular Acceleration in a Dual-Wheel Pulley System?

[royguitarboy]

Homework Statement

Two objects are attached to ropes that are attached to wheels on a common axle as shown below. The two wheels are glued together so that they form a single object. The total moment of inertia of the object is 41 kg·m2. The radii of the wheels are R1 = 1 m and R2 = 0.4 m.

(a) If m1 = 24 kg, find m2 such that there is no angular acceleration of the wheels.

(b) If 12 kg is gently added to the top of m1, find the angular acceleration of the wheels.

Homework Equations

T=I(alpha)

alpha=change in velocity/ change in time

The Attempt at a Solution

I'm not really sure where to start for this. I know I need to figure out the Tension for the block, but I'm not sure how to go about doing this.

Homework Statement

 

[Doc Al]

Start by labeling all the forces acting on each mass and on the wheel.

Hint for a: It's in equilibrium. What does that tell you about the sum of the forces or the sum of the torques?

Hint for b: Apply Newton's 2nd law to each object and combine the resulting equations.

 

[royguitarboy]

I know that the overall net Force is 0 because there is no acceleration. All I get when I apply Newton's Laws is F=T-mg. This would mean that the Tension equals mg, which I don't think is true. I think I need to figure out the relationship between the two tensions, but I'm not sure how.

 

[Doc Al]

I know that the overall net Force is 0 because there is no acceleration. All I get when I apply Newton's Laws is F=T-mg. This would mean that the Tension equals mg, which I don't think is true.

It is most definitely true. (For part a.)

I think I need to figure out the relationship between the two tensions, but I'm not sure how.

By examining the torques on the wheel. They must total zero as well.

 

[royguitarboy]

Wouldn't the tensions equaling mg mean that m2 would need to equal m1?

 

[Doc Al]

Wouldn't the tensions equaling mg mean that m2 would need to equal m1?

Not at all. Each rope has its own tension: [itex]T_1 = m_1 g[/itex] and [itex]T_2 = m_2 g[/itex]. The tension in each rope cannot be equal, since the wheel is in equilibrium. (The ropes attach to the wheel at different radii.)

 

[royguitarboy]

So how do I go about figuring out the tensions?

 

[Doc Al]

For part (a) the tensions are equal to the weights, as I thought you realized.

 

[royguitarboy]

I understand that the tensions equal mass times gravity. I need to figure out what the second mass is, and I'm pretty sure I do this by figuring out how the tensions are related.

 

[Doc Al]

Right. Set the net torque on the wheel equal to zero. That will tell you how the tensions are related.

 

[royguitarboy]

I'm sorry I'm having a hard time understanding this; The tensions must cancel each other since there is no acceleration right? Maybe if it's explained in another way I might get it.

 

[Doc Al]

The tensions must cancel each other since there is no acceleration right?

No. The torques must cancel, otherwise the wheel will accelerate. Torque depends on both force (the tension) and distance from the axis (radius):

[tex]T_1 R_1 = T_2 R_2[/tex]

 

[royguitarboy]

Ok, I have the first part now. For the second, I get that the T=m1(g+a) and T=m2(g-a), and then that a=(m1g-m2g)/(m1+m2). Is that even close for (b) If 12 kg is gently added to the top of m1, find the angular acceleration of the wheels ?

 

[Doc Al]

No, it's not that simple. The tension in each rope and the acceleration of each mass is different. Remember that you want the angular acceleration of the wheel, not the linear acceleration of the masses. How are they related?