Coupled acceleration and relative motion
| English | 中文 | Pinyin |
|---|---|---|
| constraint/kənˈstreɪnt/ | 约束 | yuē shù |
| relative velocity/ˈrelətɪv vəˈlɒsɪti/ | 相对速度 | xiāng duì sù dù |
A decision before an answer
- A load released from an aircraft keeps the aircraft’s horizontal velocity. Relative to the aircraft, its initial motion can therefore be purely downward.
- Your goal: Solve coupled-body equations using one acceleration constraint.
Read the relationship
- For a mass m1 on a frictionless horizontal table connected over an ideal massless pulley to a hanging mass m2, choose table motion and downward hanging motion as positive. A taut inextensible string gives one acceleration magnitude. Write T=m1a and m2g−T=m2a. Adding eliminates the internal tension, so a=m2g/(m1+m2) and T=m1m2g/(m1+m2). Thus T<m2g when both masses are positive. Setting tension equal to hanging weight silently assumes zero acceleration.
- Transform velocities between inertial frames.
For m1=4 kg on a frictionless table and m2=1 kg hanging, g=10 m/s², tension is:
a=10/5=2 m/s² and T=m1a=8 N.
Use the defining rule
- The common tension assumption requires an ideal massless string and pulley with negligible friction and inertia. A pulley with rotational inertia can have different tensions, with (T2−T1)R=Iα and a=Rα if the string does not slip. Draw each body separately: a force internal to the combined system remains an external force on one chosen body. Constraint forces can cancel from a system equation while still being needed for individual motion.
- Check force and energy limits for constrained motion.
A payload is dropped from a constant-speed level aircraft. With g=10 m/s² and no drag, its velocity relative to that aircraft after 2 s is:
Horizontal velocities cancel; vertical relative speed is gt=20 m/s downward.
Check the conditions
- For two inertial frames with constant relative velocity V, Galilean velocity transformation is v_relative=v_ground−V. A payload released by a level aircraft initially has the aircraft’s horizontal speed. With negligible air resistance, horizontal ground velocity remains constant and vertical velocity becomes −gt if upward is positive. Relative to that aircraft, horizontal velocity is zero and downward speed is gt. Position is a different question: vertical displacement is −½gt². Do not confuse ground speed with relative speed.
- Check force and energy limits for constrained motion.
With m1=3 kg, m2=2 kg and g=10 m/s², acceleration is 4 m/s² and tension is 12 N, below the hanging weight 20 N. From rest after a 0.5 m descent, v²=2·4·0.5=4, so v=2 m/s. A payload released by an aircraft at constant horizontal speed 80 m/s has ground velocity (80,−30) m/s after 3 s, but relative velocity (0,−30) m/s.
For m1=m2 and an ideal frictionless table/pulley, a/g=____ (decimal).
a=m2g/(m1+m2)=g/2.
Apply the task format
- Check limiting cases after solving. As m1 approaches zero, the ideal coupled acceleration approaches g and tension approaches zero; as m1 grows very large, acceleration approaches zero and tension approaches m2g from below. An energy derivation gives m2g s=½(m1+m2)v² for release from rest and an ideal pulley, agreeing with v²=2as. This agreement checks the equations; it does not permit energy conservation if friction or other unaccounted work is present.
- Check force and energy limits for constrained motion.
A massless ideal pulley equates tension, not tension and weight. Subtract frame velocities component by component before taking a speed magnitude.
Which answer fits this case?
Solve coupled-body equations using one acceleration constraint
A string constraint can make two connected bodies share acceleration magnitude while their net forces differ.
Net forces are m1a and m2a; equal acceleration does not imply equal masses or forces.
Keep the distinctions
- constraint 约束 — A condition linking allowed positions or motions of connected bodies.
- relative velocity 相对速度 — Velocity of one object measured in another moving frame.
- Solve coupled-body equations using one acceleration constraint.
- Transform velocities between inertial frames.
- Check force and energy limits for constrained motion.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.