Find angle and masse 2 of an inclined pulley system.

Question

A block of mass m1 is suspended vertically with a rope connected to a block of mass m2. The block of mass m2 is on a frictionless plane inclined at theta with respect to the horizontal. If m1=2kg, the acceleration of the block is 1.96 m/s^2 downward. If m1=1kg the acceleration of the block is 1.22 m/s^2 upwards. Determine theta (angle) and m2.

Answer 1

Hey Alara,

It helps to realize that what pulleys do is redirect a force along a rope, so we can sort of treat the forces as acting in one dimension along the rope. My professor used to call it the "sock method", because he drew a sock around the rope. We are used to finding the sum of the forces in the x or y direction, but what this method lets us do is add the forces along the direction of the rope.

So, for this problem, you have the weight of m1 pulling one way, and then you have a component of the weight of m2 pulling in the other direction. I'd call the weight of m1 positive, since that's the way the acceleration goes, and I'd call the other force negative. For the component of m2 down the plane, you should have this in your notes -- the force is mg*sin(theta).

So you have one big equation for each scenario, which comes from F = ma being applied to the system as a whole:

1. One one side, you have those 2 forces, which contain 2 unknowns (m2 and theta)
2. On the other side you have the acceleration that they give you times the total mass of both blocks, which also contains an m2.

Now you'll have a bit of algebra to do. 2 equations with 2 unknowns is a solvable system of equations that you can approach in various ways. The most straightforward way would probably be to solve one equation for m2, and then plug that expression into the other equation, which will now just have the one unknown of theta. When you do find theta, you can plug back into either equation to find m2.

Hope this helps.