Solving Blocks & Pulley Homework: Find Acceleration

[Krappy]

Homework Statement

[PLAIN]http://img109.imageshack.us/img109/1267/picture2ae.png

$$m_1 = 2kg$$
$$m_2 = 6kg$$
$$R = .25cm$$
$$M = 10kg$$
$$\theta = 30º$$
$$\mu = 0.36$$

Find the acceleration of the system.

Homework Equations

The Attempt at a Solution

$$m_1 a = T_1 - m_1 g \mu$$
$$m_2 a = m_2 g \sin \theta - m_2 g \mu \cos \theta - T_2$$
$$\tau = I a/R = R T_2 - R T_1 \cos \theta$$I solved this system and the result is different from the solution.
What am I missing/doing wrong?Regards
Johnny

 

[Doc Al]

$$\tau = I a/R = R T_2 - R T_1 \cos \theta$$

Why the cosθ factor?

 

[Krappy]

Why the cosθ factor?

Because of the cross product. |r||F| * sin(x), but in that case, sin(x) is the cos(theta).

 

[Doc Al]

Because of the cross product. |r||F| * sin(x), but in that case, sin(x) is the cos(theta).

When taking the cross product, the angle is between the vectors r and F. Since the ropes are tangential to the pulley, that angle is 90 degrees.

 

[Krappy]

I have though about this and I think that T1 must equal to T2 because, if not, the string would be extending. Am I thinking right? I ask this since before I though that it was the difference between the two tensions T1 and T2 that would accelerate the pulley, but now, I don't think that's the case.

Now I have:

$$m_1 a = T - m_1 g \mu$$
$$m_2 a = m_2 g \sin \theta - m_2 g \mu \cos \theta - T - \frac{Ia}{r^2}$$


Which one is correct?

 

[Doc Al]

I have though about this and I think that T1 must equal to T2 because, if not, the string would be extending. Am I thinking right?

No. Without a difference in string tension there would be no net torque to accelerate the pulley.

I ask this since before I though that it was the difference between the two tensions T1 and T2 that would accelerate the pulley, but now, I don't think that's the case.

You were right the first time. (Why did you change your mind?)

Now I have:

$$m_1 a = T - m_1 g \mu$$
$$m_2 a = m_2 g \sin \theta - m_2 g \mu \cos \theta - T - \frac{Ia}{r^2}$$


Which one is correct?

In addition to not reflecting the difference in tensions, the second equation looks odd since it has a term relating to the pulley. But the pulley only affects m_2 via the tension in the string that attaches to m_2.

You need three equations: one for each block and one for the pulley.

 

[Krappy]

Thank you Doc Al.

I got confused since that with the last system I got the right answer. (Notice that if I substitute the equation T2-T1 = Ia/R^2 in the original second one, I get the same thing). The thing that I was missing in the first place was that both tensions were tangential to the pulley.

Thank you very much. ;)