Category: HC Verma Part 1: Waves & Optics

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: $y=10\sin(2\pi t-0.005x+\pi/3)$ cm. Find amplitude, frequency, wave speed, initial phase. $f=\omega/2\pi$; $v=\omega/k$ Step 1: $A=10$ cm, $\omega=2\pi$ rad/s, $k=0.005$ rad/cm, $\phi_0=\pi/3$. Step 2: $f=1$ Hz; $v=\omega/k=2\pi/0.005\approx1257$ cm/s. $$\boxed{A= Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental…

Problem Statement Wire: $\mu=9.8\times10^{-3}$ kg/m, tension 10 N. Shortest length resonating at 100 Hz? 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 solving systematically with careful attention…

Problem Statement Solve the work-energy problem: Solve the work-energy problem: Standing wave $y=2a\sin(kx)\cos(\omega t)$ in string of cross-section $s$, density $\rho$. Total energy between two consecutive nodes? Total energy between nodes: (energy/length)$\times(\lambda/2)$ Step 1: Node separation $=\lambda/2$. Energy per unit length (total for Given Information $y=2a$ Physical Concepts & Formulas This problem applies fundamental physics…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: String with $\mu_0$ over frictionless pulley, mass $m$ hanging. Find wave speed on horizontal portion. $T=mg$ from hanging mass; $v=\sqrt{T/\mu}$ Step 1: Tension $T=mg$; $v=\sqrt{mg/\mu_0}$. $$\boxed{v=\sqrt{mg/\mu_0}}$$ Mass $m$ and spring constant $k$ (or equiva Given Information See problem statement for all given quantities. Physical Concepts…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Two waves each amplitude 1.0 mm, frequency 10 Hz travel in opposite directions. Antinode amplitude? Standing wave antinode amplitude $=2a$ Step 1: Antinode amplitude $=2a=2.0$ mm. $$\boxed{A_{antinode}=2.0\text{ mm}}$$ Mass $m$ and spring constant $k$ (or equivale Given Information $=2a$ $=2a$ $\boxed{A_{antinode}=2.0\text{ mm}$ Physical Concepts &…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Maximum particle speed = 4 times wave speed. Find $A/\lambda$. $v_{p,max}=A\omega=2\pi Av/\lambda$ Step 1: $v_{p,max}=A\omega=A(2\pi v/\lambda)=4v$. Step 2: $A/\lambda=4/(2\pi)=2/\pi$. $$\boxed{A/\lambda=2/\pi}$$ Mass $m$ and spring constant $k$ (or equivalent), o Given Information $v_{p,max}=A\omega=A(2\pi v/\lambda)=4v$ $A/\lambda=4/$ $\boxed{A/\lambda=2/$ Physical Concepts & Formulas This problem applies fundamental physics principles…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Wave: amplitude 0.50 mm, wavelength 1 m, frequency 2 Hz, tension 5 N. Find average power. $P=\frac{1}{2}\mu\omega^2 A^2 v$ Step 1: $v=f\lambda=2$ m/s; $\mu=T/v^2=5/4=1.25$ kg/m; $\omega=4\pi$ rad/s. Step 2: $P=\frac{1}{2}(1.25)(4\pi)^2(5\times10^{-4})^2(2)\approx4 Given Information $\mu=T/v^2=5/$ Physical Concepts & Formulas This problem applies fundamental physics principles to the…

Problem Statement Solve the kinematics problem: Solve the kinematics problem: $y=0.02\sin(2\pi x-2\pi t)$ m. Particle velocity at $x=1$ m, $t=1$ s? $v_y=\partial y/\partial t$ Step 1: $v_y=\partial y/\partial t=-0.02\times2\pi\cos(2\pi x-2\pi t)$. Step 2: At $x=1, t=1$: argument $=0$. $v_y=-0.04\pi\approx-0.126$ m/s. $$\boxed{v_y\approx-0.126\tex Given Information $x=1, t$ Physical Concepts & Formulas This problem applies fundamental physics principles…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: At $t=0$, a pulse traveling at 2 m/s in +x has shape $y=6/(3+x^2)$ m. Write the wave equation. $+x$ traveling: $y=f(x-vt)$ Step 1: Replace $x$ with $(x-vt)$ for +x travel: $y=6/[3+(x-2t)^2]$. $$\boxed{y=\dfrac{6}{3+(x-2t)^2}\text{ m}}$$ Mass $m$ and spring constan Given Information $y=6/$ $y=6/$ Physical Concepts &…

Problem Statement Solve the oscillation/wave problem: Wire: 40 cm, mass 3.2 g, tension 9.6 N, resonates at 150 Hz. Find harmonic number. All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) This problem draws on fundamental physical principles. The key is to identify which conse Given…