Pulleys are simple machines that consist of a rope that slides around a disk, called a block. Their main function is to change the direction of the tension force in a rope. The pulley systems that appear on SAT II Physics almost always consist of idealized, massless and frictionless pulleys, and idealized ropes that are massless and that don’t stretch. These somewhat unrealistic parameters mean that:
- The rope slides without any resistance over the pulley, so that the pulley changes the direction of the tension force without changing its magnitude.
- You can apply the law of conservation of energy to the system without worrying about the energy of the rope and pulley.
- You don’t have to factor in the mass of the pulley or rope when calculating the effect of a force exerted on an object attached to a pulley system.
The one exception to this rule is the occasional problem you might find regarding the torque applied to a pulley block. In such a problem, you will have to take the pulley’s mass into account. We’ll deal with this special case in Chapter 7, when we look at torque.
The Purpose of Pulleys
We use pulleys to lift objects because they reduce the amount of force we need to exert. For example, say that you are applying force F to the mass in the figure above. How does F compare to the force you would have to exert in the absence of a pulley?
To lift mass m at a constant velocity without a pulley, you would have to apply a force equal to the mass’s weight, or a force of mg upward. Using a pulley, the mass must still be lifted with a force of mg upward, but this force is distributed between the tension of the rope attached to the ceiling, T, and the tension of the rope gripped in your hand, F.
Because there are two ropes pulling the block, and hence the mass, upward, there are two equal upward forces, F and T. We know that the sum of these forces is equal to the gravitational force pulling the mass down, so F + T = 2F = mg or F = mg/2. Therefore, you need to pull with only one half the force you would have to use to lift mass m if there were no pulley.
Standard Pulley Problem
The figure above represents a pulley system where masses m and M are connected by a rope over a massless and frictionless pulley. Note that M > m and both masses are at the same height above the ground. The system is initially held at rest, and is then released. We will learn to calculate the acceleration of the masses, the velocity of mass m when it moves a distance h, and the work done by the tension force on mass m as it moves a distance h.
Before we start calculating values for acceleration, velocity, and work, let’s go through the three steps for problem solving:
- Ask yourself how the system will move: From experience, we know that the heavy mass, M, will fall, lifting the smaller mass, m. Because the masses are connected, we know that the velocity of mass m is equal in magnitude to the velocity of mass M, but opposite in direction. Likewise, the acceleration of mass m is equal in magnitude to the acceleration of mass M, but opposite in direction.
- Choose a coordinate system: Some diagrams on SAT II Physics will provide a coordinate system for you. If they don’t, choose one that will simplify your calculations. In this case, let’s follow the standard convention of saying that up is the positive y direction and down is the negative y direction.
- Draw free-body diagrams: We know that this pulley system will accelerate when released, so we shouldn’t expect the net forces acting on the bodies in the system to be zero. Your free-body diagram should end up looking something like the figure below.
