Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: A cylindrical body of mass $m$ and cross-section $S$ floats in a liquid of density $\rho$. It is pushed down slightly and released. Find the period of vertical oscillations. At equilibrium, the buoyant force equals weight: $\rho g S h_0 = mg$, where $h_0$ is the submerged depth. For…

Problem Statement Solve the oscillation/wave problem: A simple pendulum of length $l$ is in a lift (elevator) accelerating upward at $w$. Find the period. In the accelerating frame, there is a pseudo-force $mw$ downward on the bob, adding to gravity. The effective gravitational acceleration is: $$g_{\rm eff} = g + w$$ The period becomes: $$\boxed{T…

Problem Statement Solve the oscillation/wave problem: A disc of moment of inertia $I$ is suspended by a wire of torsional rigidity $k$ (restoring torque $= -k\theta$). Find the period. The equation of motion is: $$I\ddot\theta = -k\theta \implies \ddot\theta + \frac{k}{I}\theta = 0$$ $$\boxed{T = 2\pi\sqrt{\frac{I}{k}}}$$ This is analogous to the linea Given Information See…

Problem Statement Solve the oscillation/wave problem: At displacement $x_1$ the velocity is $v_1$; at displacement $x_2$ the velocity is $v_2$. Find $\omega$ and the amplitude. From $v = \omega\sqrt{a^2 – x^2}$, write two equations: $$v_1^2 = \omega^2(a^2 – x_1^2), \qquad v_2^2 = \omega^2(a^2 – x_2^2)$$ Subtracting: $$v_1^2 – v_2^2 = \omega^2(x_2^2 – x Given Information…

Problem Statement Solve the oscillation/wave problem: Find the total energy of a particle of mass $m$ performing SHM with amplitude $a$ and angular frequency $\omega$. Kinetic energy: $K = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2 a^2\sin^2(\omega t+\phi)$ Potential energy: $U = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 a^2\cos^2(\omega t+\phi)$ Total energy Given Information See problem statement for all given quantities. Physical…

Problem Statement Solve the oscillation/wave problem: A particle is subject to two SHMs in the same direction: $x_1 = a_1\cos(\omega t)$ and $x_2 = a_2\cos(\omega t + \delta)$. Find the resultant. The resultant $x = x_1 + x_2$ is also SHM at the same frequency: $$x = A\cos(\omega t + \psi)$$ where the amplitude and…

Problem Statement Solve the oscillation/wave problem: A particle executes $x = a\cos(\omega t)$ and $y = b\cos(\omega t + \delta)$. Find the trajectory for various $\delta$. Eliminate $t$: $x/a = \cos(\omega t)$ and $y/b = \cos\delta\cos(\omega t) – \sin\delta\sin(\omega t)$. $$\frac{y}{b} = \frac{x}{a}\cos\delta – \sin\delta\sqrt{1-\frac{x^2}{a^2}}$$ Given Information See problem statement for all given quantities. Physical…

Problem Statement Solve the oscillation/wave problem: In SHM, show that velocity leads displacement by $90°$. Sketch the phase diagram. With $x(t) = a\cos(\omega t + \phi)$ and $v(t) = -a\omega\sin(\omega t + \phi)$: $$v = a\omega\cos\left(\omega t + \phi + \frac{\pi}{2}\right)$$ Velocity leads displacement by exactly $90°$ (or $\pi/2$ radians). Phase Given Information See problem…

Problem Statement Solve the oscillation/wave problem: A small ball of mass $m$ rolls without slipping inside a spherical bowl of radius $R$. Find the period of small oscillations. For a solid sphere rolling without slipping, the moment of inertia is $I = \frac{2}{5}mr^2$ (ball radius $r \ll R$). The effective pendulum equation for a ball…

Problem Statement Solve the oscillation/wave problem: A body of mass $m$ is placed between two springs (constants $k_1$ and $k_2$) that are both compressed. Find the oscillation frequency. When the body is displaced by $x$ from equilibrium, both springs exert restoring forces in the same direction: $$F = -(k_1 + k_2)x$$ This is equivalent to…