Elasticity & SHM — JEE Advanced PYQs

A tank contains two immiscible liquids of densities $6\rho$ and $2\rho$. The higher density liquid is filled up to a height $L/2$ from the bottom. A thin rod of density $\rho$ and length $L$ is fully immersed and hinged at the bottom so that it can oscillate freely, as shown …

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R>r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical …

As shown in the figures, a uniform rod $O{O}^{\prime }$ of length $l$ is hinged at the point $O$ and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end $(O^{\prime })$ of the rod, as show …

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. …

A block of mass 5 kg moves along the x-direction subject to the force $F=(-20x+10)\mathrm{N}$, with the value of $x$ in metre. At time $t=0\mathrm{s}$, it is at rest at position $x=1\mathrm{m}$. The position and momentum of the block at $t=(\pi /4)\mathrm{s}$ ar …

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum …

On a frictionless horizontal plane, a bob of mass $m=0.1kg$ is attached to a spring with natural length $l_0=0.1m$. The spring constant is $k_1=0.009N{m}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016N{m}^{-1}$ when $l<l_0$. Initially the bob …

A cubical solid aluminium (bulk modulus $=-V\frac{dP}{dV}=70$ GPa) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^3{\text{ kg m}}^{-3}$ …

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area $0.5{\mathrm{cm}}^2$ and, length $\sqrt{3}\mathrm{m}$ and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires …

A steel wire of diameter $0.5mm$ and Young's modulus $2\times 10^{11}N{m}^{-2}$ carries a load of mass $M$. The length of the wire with the load is $1.0m$. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire …

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: (i) when the bl …

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\omega }_1$ and ${\omega }_2$ and have total energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$, respectively. The variations of their momenta p with positions x are shown in …

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\lambda$ are kept parallel to each other. In their resulting electric field, point charges q and -q are kept in equilibrium between them. The point c …

In plotting stress versus strain curves for the materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is(are)

During Searle's experiment, zero of the Vernier scale lies between $3.20\times 10^{-2}m$ and $3.25\times 10^{-2}m$ of the main scale. The $20^{\text{th }}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of …

One end of a horizontal thick copper wire of length $2L$ and radius $2R$ is welded to an end of another horizontal thin copper wire of length $L$ and radius $R$. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wi …

A particle of mass $m$ is attached to one end of a mass-less 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 vel …

A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc …

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase …

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase …

Steel wire of length 'L' at $40^{\circ }\mathrm{C}$ is suspended from the ceiling and then a mass 'm' is hung from its free end. The wire is cooled down from 40°C to 30°C to regain its original length 'L'. The coefficient of linear thermal expansion of the steel is …

A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, $x_1(t)=A\sin \omega t$ and $x_2(t)=A\sin \left(\omega t+\frac{2\pi }{3}\right)$. Adding a third sinusoidal displacement $x_3(t)=B\sin (\omega t+\varphi )$ brings the mas …

A wooden block performs SHM on a frictionless surface with frequency, $v_0$. The block carries a charge $+Q$ on its surface. If now a uniform electric field $\vec{E}$ is switched-on as shown, then the SHM of the block will be

When a particle of mass m moves on the x-axis in a potential of the form $V(x)=k{x}^2$, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt{\frac{m}{k}}$, as can be seen easily using dimensional analysis. However, the motion of a particle c …

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its cross-sectional area is $4.9\times 10^{-7}{m}^2$. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular fre …

The mass $M$ shown in the figure oscillates in simple harmonic motion with amplitude $A$. The amplitude of the point P is

A uniform rod of length $L$ and mass $M$ is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is ge …

The $x$ - $t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=4/3\mathrm{s}$ is

A block (B) is attached to two unstretched springs S1 and S2 with spring constants $k$ and $4k$, respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction …

A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from …

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the …

Function $\mathrm{x}=\mathrm{A}{\sin }^2\omega \mathrm{t}+\mathrm{B}{\cos }^2\omega \mathrm{t}+\mathrm{C}\sin \omega \mathrm{t}\cos \omega \mathrm{t}$ represents SHM