Connected objects are common in physics problems, from two blocks tied together to elevators, pulleys, carts, and hanging masses. These systems matter because forces on one object can affect the motion of another through ropes, strings, rods, or contact. The main goal is usually to find the acceleration of the system and the internal forces such as tension or contact force.
A clear diagram and a smart choice of system can make a complicated problem much easier.
Key Facts
- For any object or chosen system, Newton's second law is ΣF = ma.
- If objects are connected by a massless, unstretchable rope, they have the same magnitude of acceleration along the rope.
- For a massless rope over a frictionless pulley, the tension is the same throughout the rope.
- Internal forces cancel when connected objects are treated as one system, so only external forces determine the system acceleration.
- For two blocks on a frictionless horizontal surface pulled by force F, a = F/(m1 + m2).
- To find tension or contact force, isolate one object and apply ΣF = ma to that object.
Vocabulary
- System
- A system is the object or group of objects chosen for analysis in a physics problem.
- Free-body diagram
- A free-body diagram is a sketch showing all external forces acting on one object or chosen system.
- Tension
- Tension is the pulling force transmitted through a rope, string, or cable.
- Internal force
- An internal force is a force that objects within the chosen system exert on each other.
- Constraint
- A constraint is a connection or condition that links the motion of objects, such as a rope forcing equal acceleration magnitudes.
Common Mistakes to Avoid
- Including internal tension when analyzing the whole system, which is wrong because forces between objects inside the chosen system cancel in pairs.
- Assuming tension always equals the pulling force, which is wrong because tension depends on which object is isolated and on the system acceleration.
- Using different accelerations for objects connected by a taut, massless rope, which is wrong because the rope constraint makes their acceleration magnitudes match along the rope.
- Forgetting friction or using the wrong direction for friction, which is wrong because friction changes the net external force and always opposes relative motion or attempted motion.
Practice Questions
- 1 Two blocks of mass 3.0 kg and 5.0 kg are connected by a light rope on a frictionless horizontal table. A 24 N horizontal force pulls the 5.0 kg block to the right. Find the acceleration of the system and the tension in the rope.
- 2 A 2.0 kg block on a frictionless table is connected over a frictionless pulley to a hanging 3.0 kg mass. Find the acceleration of the system and the tension in the rope. Use g = 9.8 m/s².
- 3 A student analyzes two connected blocks as one system to find acceleration, then analyzes one block alone to find tension. Explain why this two-step method works and why tension appears in only the second step.