AP® Physics 1: Algebra-Based · Unit 2 · Topic 2.5
Connected Systems and Atwood Machines
Objects connected by a string move with the same speed and acceleration because the string keeps them rigidly linked, so you can treat the whole system as one mass driven by the net outside force.
A string that can't stretch forces both ends to move together: if one block speeds up, the other must speed up by exactly the same amount, in the direction the string allows. That shared acceleration is the whole reason connected systems are solvable. Instead of writing separate equations for each block with the tension as an unknown force between them, you add the masses together and look only at the forces that come from outside the system. Internal forces like tension pull equally on both blocks in opposite senses, so when you sum the whole system they cancel out of the net force calculation.
An Atwood machine is two masses hanging from a string over a pulley. The outside forces are just gravity pulling down on each mass. The heavier mass falls and the lighter one rises, so the net driving force is the weight difference, (m2 - m1)g, and the total mass being pushed around is m1 + m2. Divide one by the other and you get the acceleration, with tension nowhere in the equation. To actually find the tension, you go back and apply Newton's second law to just one block, where tension is one of only two forces acting.
This connects directly to free-body diagrams and Newton's second law from earlier in the unit: the system method is just a shortcut for adding up several free-body diagrams at once. It also sets up the next step, where friction or an incline enters the picture. A modified Atwood machine with one block sliding on a rough table still has a single shared acceleration, but now friction subtracts from the net force on that side, so the same system equation just gets an extra term.
Key terms
- System method
- Treating multiple connected objects as one combined mass so internal forces like tension cancel out.
- Atwood machine
- Two masses connected by a string over a pulley, hanging on either side.
- Tension
- The pulling force transmitted along a string, equal in magnitude at both ends of an ideal string.
- Net driving force
- The unbalanced outside force on a system, here the difference in weight between the two hanging masses.
Watch it explained
- How it worksHow does treating two hanging masses as one system make the tension drop out of the math?
- Worked exampleFind the acceleration of an Atwood machine with a 3 kg mass and a 5 kg mass.
- Worked exampleFind the string tension in an Atwood machine whose blocks accelerate at 2.45 m/s squared.
- Going furtherWhat happens to a modified Atwood machine when one block sits on a rough table instead?
Check yourself
End of topic.