How Do You Calculate Acceleration and Tension in a Three-Mass Pulley System?

[artyboy]

Homework Statement

The three blocks are attached via a massless, frictionless pulley system, as
shown. The frictionless plane is inclined at an angle  = 60 degrees. When released
from rest, the 20kg block will start to slide down the plane. Find the tension in the
string.
The image is attached.

Homework Equations

Fnet = ma

The Attempt at a Solution

For the 20 kg block I got, ma = mgsinθ - T
2kg block- ma = 2T - mg
3kg block: ma = 2T - mg
The only problem is that the acceleration of all of the masses is different. How do I find the acceleration of each? Is it a ratio of the tensions, but then the 2kg and 3kg would have the same acceleration which wouldn't make sense?

 

[collinsmark]

Homework Statement

The three blocks are attached via a massless, frictionless pulley system, as
shown. The frictionless plane is inclined at an angle = 60 degrees. When released
from rest, the 20kg block will start to slide down the plane. Find the tension in the
string.
The image is attached.

Homework Equations

Fnet = ma

The Attempt at a Solution

For the 20 kg block I got, ma = mgsinθ - T
2kg block- ma = 2T - mg
3kg block: ma = 2T - mg

I think that looks about right so far.
I'd label my a's and m's though (something like a₁, a₂, and a₃, m₁, m₂, m₃ or whatnot), to avoid confusing them with each other.

The only problem is that the acceleration of all of the masses is different. How do I find the acceleration of each? Is it a ratio of the tensions, but then the 2kg and 3kg would have the same acceleration which wouldn't make sense?

It's not a ratio of tensions, no. It's a matter of geometry. The configuration of the system.

Suppose for a moment that the 2 and 3 kg masses are held in place (not allowed to accelerate) when the system is released from rest. In this situation, the 20 kg mass won't accelerate either. [Edit: the point being that the acceleration of the 20 kg mass is dependent on the acceleration of the other two masses -- and taking this a step further the acceleration of any of the three masses is dependent on the other two.]

Now let the 3 kg mass (and 20 kg mass) move freely when the system is released from rest, only holding onto the 2 kg mass. Now the 20 kg mass and 3 kg mass can both accelerate, but they won't both accelerate at the same rate. Look at the geometry of the system and determine a relationship between the 3 kg mass' acceleration and the 20 kg mass's acceleration. Once you figure that out, hold on, because you're not quite finished with this yet.

Do the same thing except hold the 3 kG mass in place instead of the 2 kg mass.

Now get crazy and hold onto the 2 and 3 kg masses together (or one in each hand, it's up to you) and lift both masses up such that they both accelerate at the same rate, say 1 m/s². What's the acceleration of the 20 kg mass this time? You should be able to figure out a relationship (i.e. an equation) between the three accelerations. This equation is doesn't have anything to do with the tension by the way (so the equation is not going to have a T in it), it's just based on the configuration of the system.

That gives you your fourth simultaneous equation. Which is nice because you have four unknowns, a₁, a₂, and a₃ and T. Four equations, four unknowns. The rest is algebra.