Сцепленное ускорение и относительное движение
Введено| English | Значение |
|---|---|
| pulley/ˈpʊli/ | A wheel over which a rope or belt runs, used to change the direction of a force or transmit motion. |
| tension/ˈtenʃn/ | A pulling force transmitted through a stretched string, cable or similar object. |
| relative velocity/ˈrelətɪv vəˈlɒsɪti/ | относительная скорость |
Falling straight down — relative to the plane
- A payload released from a level aircraft keeps the aircraft's horizontal velocity. Seen from the aircraft, it simply drops.
- GRE mechanics tests coupled bodies through strings and pulleys 滑轮, and velocities in moving frames.
The table-and-pulley system
- $m_1$ on a smooth table, $m_2$ hanging. $T = m_1a$ and $m_2g - T = m_2a$, so $a = \dfrac{m_2g}{m_1 + m_2}$, $T = \dfrac{m_1m_2g}{m_1 + m_2}$.
- $T < m_2g$: if tension 张力 equalled the hanging weight, $m_2$ would not accelerate.
$m_1 = 3$ kg, $m_2 = 2$ kg, $g = 10$: $a = 4$ m/s², $T = 12$ N (below 20 N). After falling 0.5 m from rest: $v^2 = 2 \cdot 4 \cdot 0.5$, $v = 2$ m/s.
$m_1 = 4$ kg on the table, $m_2 = 1$ kg hanging, $g = 10$. Acceleration in m/s²?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The hanging weight $1 \times 10 = 10$ N accelerates both masses, 5 kg in total: $a = 10/5 = 2$ m/s².
Tension in N for the same system?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The string is the only horizontal force on the table mass: $T = m_1 a = 4 \times 2 = 8$ N.
In the table-and-pulley system the string tension equals the hanging weight.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
That would mean zero acceleration.
When the pulley is not ideal
- Equal tension on both sides needs a massless, frictionless pulley and string. A pulley with inertia $I$: $(T_2 - T_1)R = I\alpha$, $a = R\alpha$.
- Draw each body separately: tension is internal to the system but external to each body.
An ideal Atwood machine has 5 kg and 3 kg masses ($g = 10$). Acceleration in m/s²?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The net force $(5 - 3)g = 20$ N accelerates the total 8 kg: $a = 20/8 = 2.5$ m/s².
The pulley has rotational inertia. Compared with an ideal pulley, the acceleration of the system is
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Some of the energy goes into spinning the pulley.
Relative velocity 相对速度
- $\mathbf v_{\text{rel}} = \mathbf v_{\text{ground}} - \mathbf V_{\text{frame}}$, component by component.
- Aircraft at 80 m/s; after 3 s the payload's ground velocity is $(80, -30)$ m/s but its velocity relative to the aircraft is $(0, -30)$ m/s.
An aircraft flies level at 100 m/s. 2 s after release (no air resistance, $g = 10$), how fast is the payload moving relative to the aircraft, in m/s?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Relative velocity (0, −20).
Check the limits
- $m_1 \to 0$: $a \to g$, $T \to 0$. $m_1 \to \infty$: $a \to 0$, $T \to m_2g$.
- Energy check: $m_2gs = \tfrac12(m_1 + m_2)v^2$ agrees with $v^2 = 2as$ — only without friction.
Limiting cases
Check a = m₂g/(m₁ + m₂) at its limits.
Recap
- One string, one acceleration; tension below the hanging weight.
- Non-ideal pulleys need torque equations.
- Subtract frame velocities by components; check limiting cases.