結合加速度と相対運動
導入済み| English | 意味 |
|---|---|
| constraint/kənˈstreɪnt/ | 拘束条件 |
| relative velocity/ˈrelətɪv vəˈlɒsɪti/ | 相対速度 |
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:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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).
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
このケースに当てはまる答えはどれか？
連体方程式を解き、一つの加速度拘束条件を用いる
A string constraint can make two connected bodies share acceleration magnitude while their net forces differ.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.