# How to Calculate Pulley Systems ••• Aon_skynotlimit/iStock/GettyImages

You can calculate the force and action of pulley systems through the application of Newton's laws of motion. The second law works with force and acceleration; the third law indicates the direction of forces and how the force of tension balances the force of gravity.

## Pulleys: The Ups and Downs

A pulley is a mounted rotating wheel that has a curved convex rim with a rope, belt or chain that can move along the wheel's rim to change the direction of a pulling force. It modifies or reduces the effort needed to move heavy objects such as automobile engines and elevators. A basic pulley system has an object connected to one end while a controlling force, such as from a person's muscles or a motor, pulls from the other end. An Atwood pulley system has both ends of the pulley rope connected to objects. If the two objects have the same weight, the pulley will not move; however, a small tug on either side will move them in one direction or the other. If the loads are different the heavier one will accelerate down while the lighter load accelerates up.

## Basic Pulley System

Newton's second law, F (force) = M (mass) x A (acceleration) assumes the pulley has no friction and you ignore the pulley's mass. Newton's third law says that for every action there is an equal and opposite reaction, so the total force of the system F will equal the force in the rope or T (tension) + G (force of gravity) pulling at the load. In a basic pulley system, if you exert a force greater than the mass, your mass will accelerate up, causing the F to be negative. If the mass accelerates down, F is positive.

Calculate the tension in the rope using the following equation: T = M x A. Four example, if you are trying to find T in a basic pulley system with an attached mass of 9g accelerating upwards at 2m/s² then T = 9g x 2m/s² = 18gm/s² or 18N (newtons).

Calculate the force caused by gravity on the basic pulley system using the following equation: G = M x n (gravitational acceleration). The gravitational acceleration is a constant equal to 9.8 m/s². The mass M = 9g, so G = 9g x 9.8 m/s² = 88.2gm/s², or 88.2 newtons.

Insert the tension and gravitational force you just calculated into the original equation: -F = T + G = 18N + 88.2N = 106.2N. The force is negative because the object in the pulley system is accelerating upwards. The negative from the force is moved over to the solution so F= -106.2N.

## Atwood Pulley System

The equations, F(1) = T(1) - G(1) and F(2) = -T(2)+ G(2), assume the pulley has no friction or mass. It also assumes mass two is greater than mass one. Otherwise, switch the equations.

Calculate the tension on both sides of the pulley system using a calculator to solve the following equations: T(1) = M(1) x A(1) and T(2) = M(2) x A(2). For example, the mass of the first object equals 3g, the mass of the second object equals 6g and both sides of the rope have the same acceleration equal to 6.6m/s². In this case, T(1) = 3g x 6.6m/s² = 19.8N and T(2) = 6g x 6.6m/s² = 39.6N.

Calculate the force caused by gravity on the basic pulley system using the following equation: G(1) = M(1) x n and G(2) = M(2) x n. The gravitational acceleration n is a constant equal to 9.8 m/s². If the first mass M(1) = 3g and the second mass M(2) = 6g, then G(1) = 3g x 9.8 m/s² = 29.4N and G(2) = 6g x 9.8 m/s² = 58.8N.

Insert the tensions and gravitational forces previously calculated for both objects into the original equations. For the first object F(1) = T(1) - G(1) = 19.8N - 29.4N = -9.6N, and for the second object F(2) = -T(2) + G(2) = -39.6N + 58.8N = 19.2N. The fact that the force of the second object is greater than the first object and that the force of the first object is negative shows that the first object is accelerating upwards while the second object is moving downward.

#### Things You'll Need

• Calculator
• Weight of object or objects used in the pulley system

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