Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: A liquid of density $\rho$ partially fills a U-tube (cross-section area $S$, total liquid length $l$). Show it oscillates and find the period. Let $x$ = displacement of the liquid column from equilibrium. The restoring pressure difference is $2\rho g x$ acting on area $S$, so: $$F = -2\rho…

Problem Statement Derive the period of a simple pendulum (length $l$) for small oscillations. Find $l$ for $T=1.00$ s. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then…

Problem Statement A rigid body (mass $m$) pivots at distance $d$ from its CM. Find the period and equivalent simple-pendulum length. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of…

Problem Statement Solve the oscillation/wave problem: Mass $m$ connected to springs $k_1$, $k_2$. Find $\omega$ for (a) parallel and (b) series. (a) Parallel: same displacement, forces add: $k_{\rm eff}=k_1+k_2$ $$\boxed{\omega_\parallel = \sqrt{\frac{k_1+k_2}{m}}}$$ (b) Series: same force, extensions add: $k_{\rm eff}=k_1k_2/(k_1+k_2)$ $$\boxed{\omega Given Information See problem statement for all given quantities. Physical Concepts & Formulas This…

Problem Statement Solve the oscillation/wave problem: Find the fraction of the oscillation period a particle spends with $|x| \geq a/2$ (near a turning point). With $x = a\cos(\omega t)$: $x \geq a/2 \Rightarrow \cos(\omega t)\geq 1/2 \Rightarrow |\omega t| \leq \pi/3$. Time near positive turning point per period: $\Delta t = 2(\pi/3)/\omega$. Fraction Given Information…

Problem Statement Solve the oscillation/wave problem: Show $\langle K\rangle = \langle U\rangle = E/2$ in SHM. With $x = a\cos(\omega t+\phi)$: $$K = \tfrac{1}{2}m\omega^2 a^2\sin^2(\omega t+\phi), \quad U = \tfrac{1}{2}m\omega^2 a^2\cos^2(\omega t+\phi)$$ Time-averaging ($\langle\sin^2\rangle = \langle\cos^2\rangle = 1/2$): $$\boxed{\langle K\rangle = Given Information $\langle\sin^2\rangle = \langle\cos^2\rangle = 1/$ Physical Concepts & Formulas This problem applies…

Problem Statement Solve the oscillation/wave problem: A particle starts from equilibrium with maximum velocity. Amplitude $a$, period $T$. How long to reach $x = a/2$? Displacement from equilibrium: $x(t) = a\sin(\omega t)$, $\omega = 2\pi/T$. $$a\sin(\omega t) = \frac{a}{2} \implies \omega t = \frac{\pi}{6}$$ $$\boxed{t = \frac{T}{12}}$$ The remaining Given Information See problem statement for…

Problem Statement Solve the oscillation/wave problem: At $t=0$ a particle has displacement $x_0$ and velocity $v_0$ in SHM with frequency $\omega$. Find the amplitude. With $x = A\cos(\omega t+\phi)$, at $t=0$: $x_0 = A\cos\phi$ and $v_0 = -A\omega\sin\phi$. Squaring and adding: $$x_0^2 + \frac{v_0^2}{\omega^2} = A^2$$ $$\boxed{A = \sqrt{x_0^2 + \frac{ Given Information See problem…

Problem Statement Solve the oscillation/wave problem: A particle undergoes SHM with amplitude $a$, angular frequency $\omega$. Express $v$ as a function of $x$. Energy conservation with $E = \frac{1}{2}m\omega^2 a^2$: $$\frac{1}{2}mv^2 + \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}m\omega^2 a^2$$ $$\boxed{v = \omega\sqrt{a^2 – x^2}}$$ At $x=0$: $v=\omega a$ Given Information See problem statement for all given quantities.…

Problem Statement Solve the oscillation/wave problem: A particle performs SHM with amplitude $a$ and period $T$. Find $v_{\max}$ and $w_{\max}$. With $\omega = 2\pi/T$ and $x = a\cos(\omega t)$: $$v = -a\omega\sin(\omega t) \implies v_{\max} = a\omega = \frac{2\pi a}{T}$$ $$w = -a\omega^2\cos(\omega t) \implies w_{\max} = a\omega^2 = \frac{4\pi^2 a}{T^ Given Information See problem…