Elasticity & SHM — JEE Main PYQs

Two separate wires $A$ and $B$ are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire $B$ is 4 times that of the radius of wire $A$. The length of the wires …

A particle performs simple harmonic motion with a period of 2 second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is $\frac{1}{\mathrm{a}}\mathrm{s}$. The value of 'a' to the nearest integer is ______

Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section $1.0{\text{mm}}^2$) is $90{\text{ms}}^{-1}$. If the Young's modulus of wire is $16\times 10^{11}{\text{Nm}}^{-2}$, the extension of wire over its natural length is:

A particle of mass $m$ is fixed to one end of a light spring having force constant $k$ and unstretched length $\ell$. The other end is fixed. The system is given an angular speed $\omega$ about the fixed end of the spring such that it rotates in a circle in gravity free space. Th …

Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is $1:4$, the ratio of their diameters is

A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T . If th …

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period ${\mathrm{T}}_1$ and, T (ii) back and forth in a direction perpendicular to its plane, with a period $T_2$. The ratio $\frac{T_1}{T_2}$ will be:

A block of mass $m$ attached to a massless spring is performing oscillatory motion of amplitude 'A' on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining sys …

When a particle of mass $m$ is attached to a vertical spring of spring constant $k$ and released, its motion is described by $\mathrm{y}(\mathrm{t})={\mathrm{y}}_0{\sin }^2\omega \mathrm{t}$, where ' y ' is measured from the lower end of unstretched spring. Then $\omega$ is:

An object of mass $m$ is suspended at the end of a massless wire of length $L$ and area of cross - selection, A. Young modulus of the material of the wire is Y. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:

A cube of metal is subjected to a hydrostatic pressure 4 GPa. The percentage change in the length of the side of the cube is close to: (Given bulk modulus of metal, $\mathrm{B}=8\times 10^{10}$ Pa)

A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that $\mathrm{g}=3.1\pi {\mathrm{ms}}^{-2}$, what will be the tensile stress that would be developed in the wire?

A boy's catapult is made of rubber cord which is 42 cm long, with 6 mm diameter of cross-section and of negligible mass. The boy keeps a stone weighing 0.02 kg on it and stretches the cord by 20 cm by applying a constant force. When released, the stone flies off with a velocity o …

A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to:

Young's moduli of two wires $A$ and $B$ are in the ratio $7:4$. Wire $A$ is 2 m long and has radius $R$. Wire $B$ is 1.5 m long and has radius 2 mm. If the two wires stretch by the same length for a given load, then the value of $R$ is close to:

An electric dipole is formed by two equal and opposite charges $q$ with separation $d$. The charges have the same mass $m$. It is kept in a uniform electric field $E$. If it is slightly rotated from its equilibrium orientation, then its angular frequency $\omega$ is:

A simple pendulum oscillating in air has period T . The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}$ th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liq …

A simple pendulum of length 1 m is oscillating with an angular frequency $10\mathrm{rad}/\mathrm{s}$. The support of the pendulum starts oscillating up and down with a small angular frequency of $1\mathrm{rad}/\mathrm{s}$ and an amplitude of $10^{-2}\mathrm{m}$. The relat …

A pendulum is executing simple harmonic motion and its maximum kinetic energy is ${\mathrm{K}}_1$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is ${\mathrm{K}}_2$ then:

At $40^{\circ }\mathrm{C}$, a brass wire of 1 mm diameter is hung from the ceiling. A small mass, M is hung from the free end of the wire. When the wire is cooled down from $40^{\circ }\mathrm{C}$ to $20^{\circ }\mathrm{C}$ it regains its original length of 0.2 m . The value of M …

A person of mass $M$ is, sitting on a swing of length $L$ and swinging with an angular amplitude ${\theta }_0$. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $\ell (\ell \ll L)$, is …

The elastic limit of brass is 379 MPa. What should be the minimum diameter of a brass rod if it is to support a 400 N load without exceeding its elastic limit?

A simple pendulum of length L is placed between the plates of a parallel plate capacitor having electric field E, as shown in figure. Its bob has mass $m$ and charge $q$. The time period of the pendulum is given by

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length I and mass m . the rod is pivoted at its centre '$O$' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to ri …

A uniform cylindrical rod of length $L$ and radius $r$, is made from a material whose Young's modulus of Elasticity equals Y. When this rod is heated by temperature T and simultaneously subjected to a net longitudinal compressional force F , its length remains unchanged. The …

A spring whose unstretched length is $\ell$ has a force constant k . The spring is cut into two pieces of unstretched lengths ${\ell }_1$ and ${\ell }_2$ where, ${\ell }_1=\mathrm{n}{\ell }_2$ and n is an integer. The ratio $k_1/k_2$ of the corresponding force consta …

A load of mass M kg is suspended from a steel wire of length 2 m and radius 1.0 mm in Searle's apparatus experiment. The increase in length produced in the wire is 4.0 mm. Now the load is fully immersed in a liquid of relative density 2. The relative density of the material of lo …

A simple harmonic motion is represented by :
$\mathrm{y}=5(\sin 3\pi \mathrm{t}+\sqrt{3}\cos 3\pi \mathrm{t})\mathrm{cm}$
The amplitude and time period of the motion are :

A massless spring ( $\mathrm{k}=800\mathrm{N}/\mathrm{m}$ ), attached with a mass ( 500 g ) is completely immersed in 1 kg of water. The spring is stretched by 2 cm and released so that it starts vibrating. What would be the order of magnitude of the change in the temperature …

A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis, about $x=0$. When its potential energy (PE) equals kinetic energy (KE), the position of the particle will be

A rod of mass ' M ' and length ' 2 L ' is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of ' $m$ ' are attached at distance ' $\mathrm{L}/2$ ' from its centre on both sides, it reduces the oscillation frequency by $20\%$. The value o …

A rod of length $L$ at room temperature and uniform area of cross section $A$, is made of a metal having coefficient of linear expansion $\alpha /{}^{\circ }\mathrm{C}$. It is observed that an external compressive force F, is applied on each of its ends, prevents any chang …

Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $\ell$. The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rodmass system (see figure). Because of torsional constant k , the restoring tor …

The displacement of a damped harmonic oscillator is given by


The time taken for its amplitude of vibration to drop to half of its initial value is close to:

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass $m$ is placed on the sur …

A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of $10^{12}/\sec$. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = 108 and Avagadro number …

A particle is executing simple harmonic motion with a time period T. At time $t=0$, it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like :

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of:

A magnetic needle of magnetic moment $6.7\times 10^{-2}$ Am${}^2$ and moment of inertia $7.5\times 10^{-6}$ kg m${}^2$ is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is:

An external pressure $P$ is applied on a cube at $0^{\circ }\mathrm{C}$ so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and $\alpha$ is its coefficient of linear expansion. Suppose we want to bring the cube to its original size b …

A particle performs simple harmonic motion with amplitude A. Its speed is trebled at the instant that it is at a distance $\frac{2\mathrm{A}}{3}$ from equilibrium position. The new amplitude of the motion is :

A pendulum made of a uniform wire of cross sectional area A has time period T . When an additional mass M is added to its bob, the time period changes to ${\mathrm{T}}_{\mathrm{M}}$. If the Young's modulus of the material of the wire is Y then $\frac{1}{\mathrm{Y}}$ is equal 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 next $\tau \mathrm{s}$ it travels 2a, in same direction, then :

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M . The piston and cylinder have equal cross sectional area A . When the piston is in equilibrium, the volume of the gas is ${\mathrm{V}}_0$ and its pressure is ${\mathrm{P}}_0$. …

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

If a simple pendulum has significant amplitude (up to a factor of $1/\mathrm{e}$ of original) only in the period between $t=$ Os to $t=\tau \mathrm{s}$, then $\tau$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardat …

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.

If two springs $S_1$ and $S_2$ of force constants $k_1$ and $k_2$, respectively, are stretched by the same force, it is found tha …

Two particles are executing simple harmonic motion of the same amplitude A and frequency $\omega$ along the $x$-axis. Their mean position is separated by distance $X_0(X_0>A)$. If the maximum separation between them is $(X_0+A)$, the phase difference between their motion is …

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 $\mathrm{x},\mathrm{v}$ and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?