Elasticity & SHM — JEE Main PYQs

Using a simple pendulum experiment g is determined by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T?

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\mathrm{rad}/\mathrm{s}$. The frequency of this simple harmonic oscillator is___ Hz. $[$ take $\pi =\frac{22}{7}]$

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ______ cm. [Assume that the mass of the pendulum remains same.]

Two blocks with masses 100 g and 200 g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E. The energy stored in B, when spring constants $k_A$, $k_B$ of A and B, respectively satisfy the relation $4k_A=3k_B$, is:

The strain-stress plot for materials A, B, C and D is shown in the figure. Which material has the largest Young's modulus?

A cylindrical block of mass M and area of cross section A is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is

Two wires $A$ and $B$ made of different materials of length 6.0 cm and 5.4 cm, respectively and area of cross sections $3.0\times 10^{-5}{\mathrm{m}}^2$ and $4.5\times 10^{-5}{\mathrm{m}}^2$, respectively are stretched by the same magnitude under a given load. The r …

The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t)=\mathrm{A}\sin \omega \mathrm{t}$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t=T/2\beta$. The value of …

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of …

The equation of motion of a particle is given by $\mathrm{x}=\mathrm{a}\sin (50\mathrm{t}+\pi /3)\mathrm{cm}$. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are $\_\_\_\_$.

The increase in the pressure required to decrease the volume ($\Delta V$) of water is $6.3\times 10^7{\text{ N/m}}^2$. The percentage decrease in the volume is ______.
(Bulk modulus of water $=2.1\times 10^9{\text{ N/m}}^2$.)

A string A of length 0.314 m of Young's modulus $2\times 10^{10}{\text{ N/m}}^2$ is connected to another string B of length and Young's modulus both twice of those of A. This series combination of strings is then suspended from a rigid support and its free end is fixed to a loa …

The velocity of a particle executing simple harmonic motion along x-axis is described as $v^2=50-x^2$, where x represents displacement. If the time period of motion is $\frac{x}{7}$ s, the value of x is _____ .

A metal string A is suspended from a rigid support and its free end is attached to a block of mass M. Second block having mass 2 M is suspended at the bottom of the first block using a string B . The area of cross sections of strings A and B are same. The ratio of lengths of stri …

A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is $\_\_\_\_$ . (g is acceleration due to gravity)

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field ${\overrightarrow{\mathrm{E}}}_1={\mathrm{E}}_0\hat{\mathrm{x}}$. If another electric field …

The Young's modulus of steel wire of radius $r$ and length L is Y. If the radius r and length L of the wire are doubled then the value of Y

Two wires as shown in the figure below, made of steel and have breaking stress of $12\times 10^8$ N/m${}^2$. Area of cross-section of upper wire is $0.008$ cm${}^2$ and of lower wire is $0.004$ cm${}^2$. The maximum mass that can be added to pan without breaking any wire is ______ k …

A cube has side length 5 cm and modulus of rigidity $10^5$ N/m${}^2$. The displacement produced by a force of 10 N in the upper face of cube is ______ mm.

A copper wire of length 3 m is stretched by 3 mm by applying an external force. The volume of the wire is $600\times 10^{-6}{\mathrm{m}}^3$. The elastic potential energy stored in the wire in stretched condition would be ______ J. (Given Young modulus of copper …

A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4\times 10^8\mathrm{N}/{\mathrm{m}}^2$ and its radius is 4 mm, then maximum acceleration the lift can take is ______ $\mathrm{m}/{\mathrm{s}}^2$. (take …

The two wires A and B of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire A and wire B is 20/11. When the joined wire is kept under certain tension the elongations in the wires A and B are equal. If the length of wire A is 2 …

A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from $x=A$ to $x=\frac{A}{\sqrt{2}}$ is ______ sec.

Figure represents the extension ($\Delta l$) of a wire of length 1 meter, suspended from the ceiling of the room at one end with load W connected to the other end. If the cross-sectional area of the wire is $10^{-5}$ m${}^2$ then the Young's modulus of the wire is ______ N/ …

A spring stretches by 2 mm when it is loaded with a mass of 200 g. From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maximum energy in the spring are ______ Hz and ______ J, respectively. (Take $g=10$ m/ …

The frequency of oscillation of a mass m suspended by spring is ${\mathrm{v}}_1$. If the length of the spring is cut to half, the same mass oscillates with frequency ${\mathrm{v}}_2$. The value of $\frac{{\mathrm{v}}_2}{{\mathrm{v}}_1}$ is ____.

A uniform wire of length $l$ of weight w is suspended from the roof with a weight of W at the other end. The stress in the wire at $\frac{l}{3}$ distance from the top is $\left(\frac{\mathrm{W}}{\mathrm{A}}+\frac{2}{\gamma }\frac{\mathrm{w}}{\mathrm{A}}\right)$, where, A is the c …

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on …

A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $\mathrm{y}\pi \times 10^{-2}\mathrm{s}$, where the value of $y$ is
(Acceleration due to gr …

Two bodies A and B of equal mass are suspended from two massless springs of spring constant ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity …

An electric dipole of mass m , charge q , and length $l$ is placed in a uniform electric field $\overrightarrow{\mathrm{E}}={\mathrm{E}}_0\hat{\mathrm{i}}$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will b …

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.
Reason (R) : Time period of a simple pendulum decr …

Given below are two statements. One is labelled as
Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position ${\mathrm{x}}_0$ and initial momentum ${\mathrm{p}}_0$ is enough to determine the position and momentum at any time t for a simple ha …

The volume contraction of a solid copper cube of edge length 10 cm , when subjected to a hydraulic pressure of $7\times 10^6\mathrm{Pa}$, would be ______ ${\mathrm{mm}}^3$.
(Given bulk modulus of copper $=1.4\times 10^{11}{\mathrm{Nm}}^{-2}$ )

The increase in pressure required to decrease the volume of a water sample by $0.2\%$ is $\mathrm{P}\times 10^5{\mathrm{Nm}}^{-2}$. Bulk modulus of water is $2.15\times 10^9{\mathrm{Nm}}^{-2}$. The value of P is ______ .

A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then $\frac{\mathrm{D}}{\mathrm{d}}$ is :-

The length of a light string is 1.4 m when the tension on it is 5 N. If the tension increases to 7 N, the length of the string is 1.56 m. The original length of the string is ______ m.

Two blocks of masses $m$ and $M,(M>m)$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
( $\mu =$ coefficient of friction between the two blocks) …

Two simple pendulums having lengths $l_1$ and $l_2$ with negligible string mass undergo angular displacements ${\theta }_1$ and ${\theta }_2$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

Two slabs with square cross section of different materials $(1,2)$ with equal sides $(l)$ and thickness $d_1$ and $d_2$ such that $d_2=2d_1$ and $l>d_2$. Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. T …

Two wires A and B are made of same material having ratio of lengths $\frac{L_A}{L_B}=\frac{1}{3}$ and their diameters ratio $\frac{d_A}{d_B}=2$. If both the wires are stretched using same force, what would be the ratio of their respective elongations?

A 3 m long wire of radius 3 mm shows an extension of 0.1 mm when loaded vertically by a mass of 50 kg in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is $\mathrm{P}\times 10^{11}{\mathrm{Nm}}^{-2}$, where the value …

A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of $0.8{\mathrm{cm}}^3$. If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take $1\mathrm{atm}=10^5\mathrm{Pa}$)

A cylindrical rod of length 1 m and radius 4 cm is mounted vertically. It is subjected to a shear force of $10^5\mathrm{N}$ at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement $\theta$ of the rod axis from its original po …

A steel wire of length 2 m and Young's modulus $2.0\times 10^{11}{\mathrm{Nm}}^{-2}$ is stretched by a force. If Poisson ratio and transverse strain for the wire are 0.2 and $10^{-3}$ respectively, then the elastic potential energy density of the wire is _______ …

A particle oscillates along the x-axis according to x(t) = x₀ sin²(πt/2), where x₀ = 1 m. The kinetic energy (K) of the particle as a function of x is correctly represented by the graph: [graph options in source]

A wire of cross sectional area A, modulus of elasticity $2\times 10^{11}{\mathrm{Nm}}^{-2}$ and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle it sags lower from its original position making angle $\theta =\frac{1}{100}$ …

In simple harmonic motion, the total mechanical energy of given system is E . If mass of oscillating particle P is doubled then the new energy of the system for same amplitude is :

The displacement of a particle executing SHM is given by $x=10\sin \left(\omega t+\frac{\pi }{3}\right)$ m. The time period of motion is 3.14 s . The velocity of the particle at $t=0$ is ________ $\text{m/s}$.

An elastic spring under tension of 3 N has a length a. Its length is b under tension 2 N. For its length ($3a-2b$), the value of tension will be ______ N.