How to Calculate Pulley Systems

Updated July 20, 2017

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. A pulley modifies or reduces the effort to move heavy objects such as an elevator. A basic pulley system has an object connected to one end while a person controls the other end. An Atwood pulley system has both ends of the pulley rope connected to objects. If the masses of the two objects are the same weight, the pulley will not move. If the loads are different the heavier load will accelerate down while the lighter load accelerates up. The total force exerted by a pulley system can be calculated using Newton's laws of motion.

Write down the following equation: F (force) = M (mass) x A (acceleration), which is given by Newton's second law assuming there is no friction and the pulley's mass is neglected. 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 will be positive.

Calculate the tension in the rope with the calculator 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 than 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.

Write down the following equations: F(1) = T(1) - G(1) and F(2) = -T(2)+ G(2) which assumes there is no friction and the pulley's mass is neglected. This will be the case if mass two is greater than mass one. The equations would be switch if mass one was greater than mass two.

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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About the Author

Angel Coswell started her public writing career in 2008. She has excellent experience writing with eHow about home improvement, mathematical and scientific concepts, help related topics and financing. She received her master's degree in physics with an emphasis in engineering in 2008 from the University of Utah.