Pulley system problems connect forces, motion, and constraints in a way that appears often in high school physics. This cheat sheet helps students organize free-body diagrams, choose positive directions, and write Newton’s second law for connected masses. It is especially useful for Atwood machines, blocks on tables, hanging masses, and simple movable pulleys.
Worked problem patterns make it easier to recognize which forces belong on each object.
Key Facts
- For an ideal massless rope over a frictionless pulley, the tension is the same everywhere in that continuous rope, so .
- Newton’s second law for each object is , using signs based on the positive direction chosen for that object.
- For a two-mass Atwood machine with , the acceleration is .
- For a two-mass Atwood machine, the rope tension is when the pulley and rope are ideal.
- For a block on a frictionless table connected to a hanging mass , the acceleration is .
- If kinetic friction acts on a table block, the friction force is .
- For a movable pulley supporting a load with two rope segments, the upward support force is and the ideal mechanical advantage is .
- A single rope constraint means connected objects have equal acceleration magnitudes unless a movable pulley changes the distance relationship.
Vocabulary
- Tension
- Tension is the pulling force transmitted through a rope, cord, or string, usually represented by .
- Atwood machine
- An Atwood machine is a pulley system with two masses connected by a rope over a pulley.
- Free-body diagram
- A free-body diagram shows one object isolated with all external forces drawn as labeled arrows.
- Ideal pulley
- An ideal pulley is massless and frictionless, so it changes the direction of tension without changing its magnitude.
- Mechanical advantage
- Mechanical advantage is the factor by which a machine multiplies force, such as .
- Constraint equation
- A constraint equation relates the motion of connected objects because the rope length stays fixed.
Common Mistakes to Avoid
- Using one force equation for the whole system when tension is needed, because internal tension cancels only when analyzing the complete system.
- Assuming tension always equals weight, because only applies when a hanging mass has zero acceleration and no other vertical forces.
- Giving connected masses different acceleration magnitudes in a single fixed-pulley rope, because an inextensible rope forces equal magnitude motion along the rope.
- Forgetting signs in , because the direction chosen as positive determines whether forces such as or are positive or negative.
- Treating a movable pulley like a fixed pulley, because a movable pulley can have two supporting rope segments and may require a different acceleration constraint.
Practice Questions
- 1 An Atwood machine has and . Find the acceleration of the system using .
- 2 A block on a frictionless table is connected over a pulley to a hanging mass. Find the acceleration and tension.
- 3 A block on a table has and is connected to a hanging mass. Find and the system acceleration.
- 4 In an ideal movable pulley lifting a load, explain why the load can be supported by even though the person pulls with only one tension force .
Understanding Pulley System Problems Worked Examples
The most useful habit in pulley problems is to treat each moving object as its own physics problem before linking the answers. Isolate one mass at a time. Draw only the forces that actually act on it.
A hanging mass has its weight downward and the rope pull upward. A block on a surface has weight, a support force from the surface, the rope pull, and possibly friction. Do not place forces from a different object on that diagram.
Tension pulls away from the object along the rope. It never pushes. This simple rule prevents many incorrect diagrams.
Signs cause more errors than algebra. Choose a positive direction for each object that matches its expected motion. For a heavier hanging mass, down can be positive for that mass.
The lighter mass then moves up, so up can be positive on its separate diagram. Both objects can have a positive acceleration in their own equations because their chosen positive directions differ. If the final acceleration is negative, the calculation is not broken.
It means the real motion is opposite to the direction you assumed. Keep the sign choices visible until the end instead of changing them partway through.
The rope creates a geometric rule, not just a pulling force. A fixed pulley mainly changes direction, so when one end of a taut rope moves a certain distance, the other end moves the same distance in the opposite direction. A movable pulley shares the load across multiple rope sections.
If two sections support the moving pulley, a change in rope length is split between them. The load therefore moves only half as far as the free end of the rope, and its acceleration has half the magnitude.
This is why movable pulley problems cannot use the same acceleration relationship as a basic two mass setup. Sketching the rope path and marking its changing lengths is often clearer than trying to memorize a rule.
Real pulley systems are not ideal. Ropes have mass, pulleys have axle friction, and a pulley wheel needs some force to start rotating. These effects can make tensions on opposite sides slightly different and reduce the acceleration from the ideal prediction.
In school problems, ideal assumptions are usually stated or implied so that the main focus stays on force analysis. In everyday lifting equipment, the tradeoff remains important. A system that reduces the force needed usually requires pulling more rope and takes longer to raise the load.
Check units at the end. Forces should be in newtons, mass in kilograms, and acceleration in meters per second squared. An acceleration larger than gravitational acceleration in a simple gravity driven system is a warning that a force, sign, or constraint was handled incorrectly.