Solving a pulley system (physics)

Pulleys are elementary machines comprises a rope that slides around a disk which is called a block. The main function is to change the direction of the tension force in a rope. Pulleys are useful for us to lift objects as they reduce the amount of force we need to exert.

Applying force F to the mass in a pulley system will be much lesser than if compared to the force we need to exert if without the pulley.

Generally pulley systems on our physics papers are very idealized frictionless pulleys which seemed without mass. The ropes of these pulleys are also very idealized ropes which are without mass and also do not seemed to stretch. Of course in the real world, those are unrealistic as there is always energy loss along the way and that the masses of the parts in the pulley system ( i.e the rope and the blocks) are not factored in the calculation when we consider the forces exerted on the attached object.

Diagram below shows a pulley system with a load,m of 1 kg on one side and another load, M of 2 kg hanging on the other end of the rope.

Question:

If the load is released, what will be the acceleration of the system?

Explanation:

In answering the question you will need to consider how the system will go upon releasing load. Of course, using common sense we'd think that the 2 kg mass will fall and hence lifting the 1 kg mass.

As both are connected by the rope, we know that the magnitude of the velocity of 1 kg mass will be equal to velocity of 2 kg mass. Note that velocity is a vector quantity. Thus, one of them shall be taken as negative as they are moving in opposite directions.

Let's name the direction as x for 1 kg mass and -x for 2 kg mass.

Draw a free body diagram so you know what forces are working within the pulley system

The tension force which is indicated as, T, in the diagram above which occured on each of the masses is equivalent in magnitude on both side.

The tension, the velocity and acceleration, is the equal at every point in the system.

The weight of each load is shown in red arrow indicated by mg and Mg as well.

So, what is acceleration of the system?

Using Newtons Second Law:


For Mass 2 kg:  
T-Mg = Ma
     T  = 2x(-a) + 2(10)
         = -2a +20     ------(1)

For mass 1 kg:
T-mg = ma
     T  = 1x(a) + 1(10)
         = a + 10     ------(2)

  a+10 = -2a + 20
    3a  =  10
         =   3.3 m/s2



The system will accelerate at 3.3m/s2