Elasticity & SHM — JEE Main PYQs

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as s …

If $Y,K$ and $\eta$ are the value of Young's modulus, bulk modulus of rigidity of any material respectively. Choose the correct relation for these parameters

If the length of the pendulum in pendulum clock increases by $0.1\%$, then the error in time per day is

Two simple harmonic motions are represented by the equations $x_1=5\sin \left(2\pi t+\frac{\pi }{4}\right)$ and $x_2=5\sqrt{2}(\sin 2\pi t+\cos 2\pi t)$. The amplitude of second motion is _____ times the amplitude in first motion.

The value of tension in a long thin metal wire has been changed from $T_1$ to $T_2$. The lengths of the metal wire at two different values of tension $T_1$ and $T_2$ are ${\ell }_1$ and ${\ell }_2$ respectively. The actual length of the metal wire is :

A body having specific charge $8\mu \mathrm{C}/\mathrm{g}$ is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of $100\mathrm{V}/\mathrm{m}$ is applied horizontal …

Two electrons each are fixed at a distance ' $2d$ '. A third charge proton placed at the midpoint is displaced slightly by a distance $x(x\ll d)$ perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency:
(m = m …

In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant k, the frequency of oscillation of given body is:

When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is:

A uniform metallic wire is elongated by 0.04 m when subjected to a linear force $F$. The elongation, if its length and diameter is doubled and subjected to the same force will be ____ cm.

The length of a metal wire is ${\ell }_1$, when the tension in it is $T_1$ and is ${\ell }_2$ when the tension is $T_2$. The natural length of the wire is :

A particle is making simple harmonic motion along the X-axis. If at a distances $x_1$ and $x_2$ from the mean position the velocities of the particle are $v_1$ and $v_2$ respectively. The time period of its oscillation is given as :

Two simple harmonic motion, are represented by the equations $y_1=10\sin \left(3\pi t+\frac{\pi }{3}\right)$ ${\mathrm{y}}_2=5(\sin 3\pi \mathrm{t}+\sqrt{3}\cos 3\pi \mathrm{t})$
Ratio of amplitude of $y_1$ to $y_2=x:1$. The value of $x$ is ____

Wires ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ are made of same material having the breaking stress of $1.25\times 10^9\mathrm{N}/{\mathrm{m}}^2$. ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ have cross-sectional area of $8\times 10^{-7}{\mathrm{m}}^2$ and …

If the time period of a two meter long simple pendulum is $2s$, the acceleration due to gravity at the place where pendulum is executing S.H.M. is :

${\mathrm{T}}_0$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, the modified time period is :

The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by $x(t)=A\sin \omega t+B\cos \omega t$ with $\omega =\sqrt{\frac{K}{m}}$. Suppose that at time $t=0$, the position of mass is $x(0)$ and velocity $v(0)$, then its dis …

The area of cross - section of a railway track is $0.01{\mathrm{m}}^2$. The temperature variation is $10^{\circ }\mathrm{C}$, Coefficient of liner expansion of material of track is $10^{-5}/{}^{\circ }\mathrm{C}$. The energy stored per meter in the track is ______ …

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance ( $\mathrm{R}/2$ ) from the earth's centre, where ' $R$ ' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simp …

If two similar spring each of spring constant ${\mathrm{K}}_1$ are joined in series, the new spring constant and time period would be changed by a factor :

The normal density of a material its $\rho$ and its bulk modulus of elasticity is K . The magnitude of increase in density of material, when a pressure P is applied uniformly on all sides, will be :

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure. The potential energy $\mathrm{U}(\mathrm{x})$ versus time (t) plot of the particle is correctly shown in figure :

The length of metallic wire is ${\ell }_1$ when tension in it is $T_1$. It is ${\ell }_2$ when the tension is $T_2$. The original length of the wire will be :

A particle executes S.H.M., the graph of velocity as a function of displacement is:

Given below are two statements:
Statement I: A second's pendulum has a time period of 1 second.
Statement II: It takes precisely one second to move between the two extreme positions.
In the light of the above statements, choose the correct answer from the options given below:

Time period of a simple pendulum is T . The time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{\alpha }{\beta }\mathrm{T}$. The value of $\alpha$ is ____

A particle executes S.H.M. with amplitude ' $a$ ' and time period ' $T$ '. The displacement of the particle when its speed is half of maximum speed is $\frac{\sqrt{x}a}{2}$. The value of $x$ is ____

For a body executing S.H.M.:
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time interval are always equal.
(c) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E. in one time period …

Four identical hollow cylindrical columns of mild steel support a big structure of mass $50\times 10^3\mathrm{kg}$, The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each col …

A bob of mass ' $m$ ' suspended by a thread of length I undergoes simple harmonic oscillations with time period $T$. If the bob is immersed in a liquid that has density $\frac{1}{4}$ times that of the bob and the length of the thread is increased by $1/3^{\text{rd }}$ …

A uniform heavy rod of weight $10\mathrm{kg}{\mathrm{ms}}^{-2}$, cross-sectional area $100{\mathrm{cm}}^2\mathrm{f}$ and length 20 cm is hanging from a fixed support. Young modulus of the material of the rod is $2\times 10^{11}{\mathrm{Nm}}^{-2}$. Neglecting the latera …

When a rubber ball is taken to a depth of _____ m in deep sea, its volume decreases by $0.5\%$.
(The bulk modulus of rubber $=9.8\times 10^8{\mathrm{Nm}}^{-2}$
Density of sea water $=10^3{\mathrm{kgm}}^{-3}$
$g=9.8{m/s}^2$)

A particle of mass 1 kg is hanging from a spring of force constant $100{\mathrm{Nm}}^{-1}$. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system wi …

A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is $\frac{3E}{4}$ then its displacement '$y$' is given by:

A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $10^{-6}{\mathrm{m}}^2$ stretched by an amount 0.04 m. The velocity of the projected stone is ______ $\mathrm{m}/\mathrm{s}$.
(Young's modulus of rubber …

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

Consider two identical springs each of spring constant k and negligible mass compared to the mass $M$ as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{T_b}{T_a}=\sqrt{x}$, where …

Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are $Y_1$ and $Y_2$. The combination behaves as a single wire then its Young's modulus is:

In the reported figure, two bodies A and B of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will …

A particle of mass 1 mg and charge q is lying at the mid-point of two stationary particles kept at a distance '2 m' when each is carrying same change 'q'. If the free charged particle is displaced from its equilibrium position through distance '$x$' ($x\ll 1\mathrm{m}$). The pa …

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

An object of mass 0.5 kg executing simple harmonic motion. Its amplitude is 5 cm and time period $(T)$ is 0.2 s. What will be the potential energy of the object at an instant $t=\frac{T}{4}$ s starting from mean position. Assume that the initial phase of the oscillation is zero.

A particle executes simple harmonic motion represented by displacement function as $x(t)=A\sin (\omega t+\varphi )$
If the position and velocity of the particle at $\mathrm{t}=0\mathrm{s}$ are 2 cm and $2\omega \mathrm{cm}{\mathrm{s}}^{-1}$ respectively, then its amplitude is …

A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravi …

Time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration $\mathrm{g}/2$, the time period of pendulum will be:

Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass $=500\mathrm{g}$, Decay constant $=20\mathrm{g}/\mathrm{s}$ then how much time is required for the amplitude of the system to drop to half of its initial value?
( …

The function of time representing a simple harmonic motion with time period of $\frac{\pi }{\omega }$ is :

An object is located at 2 km beneath the surface of the water. If the fractional compression $\frac{\Delta \mathrm{V}}{\mathrm{V}}$ is $1.36\%$, the ratio of hydraulic stress to the corresponding hydraulic strain will be
[Given: density of water is $1000{\mathrm{kgm}}^{-3}$ and …

A block of mass 1 kg attached to a spring is made to oscillate with initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion. (take $\ln 2=0.693$)

Two particles $A$ and $B$ of equal masses are suspended from two massless springs of spring constants ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of $A$ and $B$ is