02 2-2 (b) Two masses hanging over a smooth pulley
Results
- Let hang at the ends of the string. Equations: and .
- If the masses start level, after time each has moved in opposite directions, so they are apart vertically.
The pulley reaction
- The string pulls the pulley with on each side, both downward, so the force on the pulley's axle is
- This is the pressure the pulley exerts on its support (the support's reaction is equal and opposite).
- A light pulley's own weight is ignored.
✍️ Worked examples
Example 1:
- Masses 105 g and 70 g hang over a smooth pulley, level and at rest.
- Find the acceleration.
- If the heavier body hits the ground after falling 50 cm, find the total time until the lighter body first comes to rest instantaneously.
Solution:
- cm/s.
- When the heavier mass lands: , so cm/s, after s.
- The lighter mass then rises freely with initial speed 140 cm/s and deceleration : it stops after s.
- Total time s.
Example 2:
- A 90 g mass and a 70 g mass hang from a string over a smooth pulley.
- The 90 g mass starts from rest 245 cm above the ground.
- Find (a) the time for it to reach the ground and (b) the time until the string becomes taut again.
Solution:
- cm/s.
- (a) , so and s. Its speed on landing is cm/s.
- (b) The 70 g mass is then 245 cm above its start, moving up at 245 cm/s. The string goes slack and the mass moves freely under gravity.
- The string becomes taut when the 70 g mass has returned to the height of 245 cm, i.e. after s.
- That is 2.5 s from the beginning of the motion.
Example 3: A 5 kg and a 3 kg mass hang level over a smooth pulley. Find , and the force on the pulley.
Solution:
- m/s.
- N.
- Force on the pulley: N.
Example 4: Two masses start level. After 1 s the vertical distance between them is 20 cm. Find .
Solution:
- Each has moved 10 cm: , so cm/s.
- , so .
- , so .