Embibe Experts Solutions for Chapter: Simple Harmonic Motion, Exercise 3: Exercise - 3

A wooden cube (density of wood $d$) of side $l$ floats in a liquid of density $\mathrm{\rho }$ with its upper and lower surfaces horizontal. If the cube is pushed slightly down and released, it performs simple harmonic motion of period $T$ then $T$ is equal to

A $0.1 \mathrm{kg}$ mass is suspended from a wire of negligible mass. The length of the wire is $1 \mathrm{m}$ and its cross-sectional area is $4.9\times {10}^{-7} {\mathrm{m}}^2$. If the mass is pulled a little in the vertically downward direction and released, it performs a simple harmonic motion of angular frequency $140 \mathrm{rad} {\mathrm{s}}^{-1}$. If Young's modulus of the material of the wire is $n\times {10}^9 \mathrm{N} {\mathrm{m}}^{-2}$, the value of $n$ is

A point mass is subjected to two simultaneous sinusoidal displacements in X-direction $x_1\left(t\right)=A\sin \omega t$ and $x_2\left(t\right)=A\sin (\omega t+\frac{2\pi }{3})$. Adding a third sinusoidal displacement $x_3\left(t\right)=B\mathit{ }\sin (\omega t+\varphi )$ brings the mass to a complete rest. The values of $B$ and $\varphi$ are:

A particle of mass $m$ is attached to one end of a massless spring of force constant $k$ lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5$$u_0$, it collides elastically with a rigid wall. After this collision,

A mass $M$, attached to a horizontal spring, executes s.h.m. with amplitude $A_1$. When the mass $M$ passes through its mean position then a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ is
 

If a simple pendulum has significant amplitude (up to a factor of $\frac{1}{e}$ of original) only in the period between $t=0$ to$\mathit{ }t=\mathrm{\tau }$, then $\mathrm{\tau }$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with $b$ as the constant proportionality, the average lifetime of the pendulum in (assuming damping is small) in seconds$:$

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude is $5 \mathrm{s}$. In another $10 \mathrm{s}$, it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals to

A particle moves with simple harmonic motion in a straight line. In first $\tau \mathrm{s}$ after starting from rest it travels a distance $a$ and in the next $\tau \mathrm{s}$ it travels $2a$ in the same direction, then