Oscillations L6 Block–Spring Collisions, Impulse in SHM, Elastic Wall, Two Body SHM & Reduced Mass скачать в хорошем качестве

Oscillations L6 Block–Spring Collisions, Impulse in SHM, Elastic Wall, Two Body SHM & Reduced Mass

In Oscillations Lecture 6, we go beyond the basic spring–mass system and study non‑standard SHM situations and two‑body SHM. We connect these problems directly to the System of Particles / COM chapter using the ideas of COM and reduced mass. What we cover in this lecture: Block hitting a spring (not attached) Block of mass (m) with speed (v_0) hits a light spring of constant (k), compresses it and then leaves: While in contact, motion is SHM with [ \omega = \sqrt{\frac{k}{m}}. ] Using energy conservation: [ \frac12 mv_0^2 = \frac12 kA^2 \Rightarrow \boxed{A = v_0\sqrt{\frac{m}{k}}}. ] Time from first contact to leaving the spring: [ t_{\text{contact}} = \pi\sqrt{\frac{m}{k}} ] (either from the SHM equation or using the “mean to extreme = (T/4)” shortcut). Block moving between two springs (oscillatory but not SHM) Block moves between two separated springs (k_1) and (k_2) on a smooth table: In contact with each spring: half‑cycle SHM with [ t_1 = \pi\sqrt{\frac{m}{k_1}},\quad t_2 = \pi\sqrt{\frac{m}{k_2}}. ] Between the springs (C to D): both springs relaxed, no horizontal force, [ a = 0 \Rightarrow v = \text{constant},\quad t_3 = \frac{2L}{v}. ] Total time for one “oscillation”: [ T = t_1 + t_2 + t_3. ] Since between C and D, (F=0), the motion is oscillatory but not purely SHM (does not satisfy (F=-kx) everywhere). Impulse during SHM (changing amplitude and phase) Particle in SHM ((k, m), amplitude (A)) at (y = +A/2) receives an impulse that doubles its velocity: Before impulse: [ v = \omega\sqrt{A^2 - (A/2)^2} = \frac{\sqrt{3}}{2},\omega A. ] After impulse: [ v' = 2v = \sqrt{3},\omega A. ] New amplitude (A_1) from [ v' = \omega\sqrt{A_1^2 - (A/2)^2} \Rightarrow \boxed{A_1 = \frac{\sqrt{13}}{2}A}. ] Time to reach the right extreme after impulse using phasor: [ \cos\theta = \frac{y}{A_1} = \frac{1}{\sqrt{13}}, \quad t = \frac{\theta}{\omega} = \sqrt{\frac{m}{k}}\cos^{-1}!\left(\frac{1}{\sqrt{13}}\right). ] Impulse changes amplitude and phase, but not (\omega) (since (m, k) unchanged). Block–spring with an elastic wall (modified period) Block attached to spring ((k, m)) with amplitude (A). An elastic wall is placed at (y = A/2) on one side: Block collides elastically with wall and reverses direction without energy loss. On phasor circle: [ \cos\theta = \frac{A/2}{A} = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{3}. ] Collision causes a phase jump of (2\theta = \frac{2\pi}{3}). Time for this phase change: [ \Delta t = \frac{T}{2\pi}\cdot\frac{2\pi}{3} = \frac{T}{3}. ] Effective period of constrained motion: [ T' = T - \frac{T}{3} = \frac{2T}{3} = \frac{2}{3}\left(2\pi\sqrt{\frac{m}{k}}\right) = \boxed{\frac{4\pi}{3}\sqrt{\frac{m}{k}}}. ] Reminder: COM and reduced mass from System of Particles No external horizontal force ⇒ COM velocity constant. If COM is at rest: [ m_1 x_1 + m_2 x_2 = 0 \Rightarrow m_1 x_1 = m_2 x_2. ] For internal (relative) motion of two masses: [ \boxed{\mu = \frac{m_1 m_2}{m_1 + m_2}} ] behaves like a single “effective mass” attached to the spring. Two‑body SHM: two blocks connected by a spring (reduced mass) Two masses (m_1, m_2) connected by spring (k), on a smooth surface, with initial compression (x_0): Displacements from equilibrium: (x_1, x_2). With COM at rest: [ m_1 x_1 = m_2 x_2. ] Spring extension/compression: [ \Delta\ell = x_1 + x_2. ] Equation of motion for one mass leads to: [ \omega^2 = \frac{k(m_1 + m_2)}{m_1 m_2} = \frac{k}{\mu}, \quad \boxed{T = 2\pi\sqrt{\frac{\mu}{k}}}. ] Amplitudes from COM condition and geometry: [ A_1 + A_2 = x_0,\quad m_1 A_1 = m_2 A_2 ] giving [ \boxed{ A_1 = \frac{m_2}{m_1 + m_2}x_0,\quad A_2 = \frac{m_1}{m_1 + m_2}x_0. } ] Example (3 kg & 6 kg blocks, spring 800 N/m, 6 cm compression) Reduced mass: [ \mu = \frac{3\times 6}{3+6} = 2\ \text{kg}. ] Time period: [ T = 2\pi\sqrt{\frac{2}{800}} = \frac{\pi}{10}\ \text{s}. ] Amplitude of 3 kg block: [ A_1 = \frac{6}{9}\times 6\ \text{cm} = 4\ \text{cm}. ] Using momentum and energy conservation at mean position: [ p_{2,\max} = 2.4\ \text{kg m/s}. ] This lecture shows how System of Particles/COM ideas (COM, reduced mass, relative motion) are directly used in SHM problems with two bodies and constraints.