Simple Harmonic Motion (SHM)

Given an one dimensional system with total energy $E=\frac{{p_x}^2}{2m}+V\left(x\right)=$ constant, where $p_x$ is the $x$ component of the linear momentum and $V\left(x\right)$ is the potential energy of the system. Show that total time derivative of energy gives us force $F_x=-\frac{d}{dx}V\left(x\right)$. Verify Hooke’s law by choosing potential energy $V\left(x\right)=\frac{1}{2}k{x}^2$.

A mass of $3 \mathrm{kg}$ is attached at the end of a spring moves with simple harmonic motion on a horizontal frictionless table with time period $2\pi$ and with amplitude of $2 \mathrm{m}$, then the maximum fore exerted on the spring is

A particle executes simple harmonic motion and displacement $y$ at time $t_0, 2t_0$ and $3t_0$ are $A, B$ and $C$, respectively. Then the value of $\left(\frac{A+C}{2B}\right)$ is

Let the total energy of a particle executing simple harmonic motion with angular frequency is $1 \mathrm{rad} {\mathrm{s}}^{-1}$ is $0.256 \mathrm{J}$. If the displacement of the particle at time $\mathrm{t}=\frac{\mathrm{\pi }}{2} \mathrm{s}$ is $8\sqrt{2} \mathrm{cm}$ then the amplitude of motion is

The displacement of a simple harmonic motion is given by $y\left(t\right)=A \sin (\omega t+\phi )$ where $A$ is amplitude of the oscillation, ($\omega$ is the angular frequency and $\phi$ is the phase. Let the amplitude of the oscillation be $8 \mathrm{cm}$ and the time period of the oscillation is $24 \mathrm{s}$. If the displacement at initial time $(\mathrm{t}=0 \mathrm{s})$ is $4 \mathrm{cm}$, then the displacement at $t=6 \mathrm{s}$ is

In a simple harmonic oscillation, the shape of acceleration against displacement for one complete oscillation will be

Write down the time period of a simple pendulum.

State the laws of the simple pendulum.

What is an epoch?

Define frequency of simple harmonic motion.

Define the time period of simple harmonic motion.

Consider two simple harmonic motion along x and y axis having same frequencies but different amplitudes as $x=A\sin (\omega t+\varphi )$ (along x-axis) and $y=B\sin \omega t$ (along y-axis).Then show that $\frac{x^2}{A^2}+\frac{y^2}{B^2}-\frac{2xy}{AB}\cos \varphi ={\sin }^2\varphi$.

And also discuss the special cases when $\varphi =0; \varphi =\mathrm{\pi }; \varphi =\frac{\mathrm{\pi }}{2};$$\varphi =\frac{\mathrm{\pi }}{2} \mathrm{and} A=B; \varphi =\frac{\mathrm{\pi }}{4}$.

A piece of wood of mass $m$ is floating erect in a liquid whose density is $\rho$. If it is slightly pressed down and released, then executes simple harmonic motion. Show that its time period of oscillation is $T=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{Ag\rho }}}$.

Describe simple harmonic motion as projection of uniform circular motion.

What is meant by simple harmonic oscillation? Give examples and explain why every simple harmonic motion is a periodic motion whereas the converse need not be true.