Oscillations

Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to A and T, respectively. At time t = 0 one particle has displacement A while the other one has displacement $\dfrac{-A}2$ and they are moving towards each other. If they cross each other at time t, then t is :

A simple pendulum made of a bob of mass m and a metallic wire of negligible mass has time period 2 s at T = 0°C. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope S. If the coefficient of linear expansion of metal is α then the value of S is :

In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to :

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $\omega_1$ and $\omega_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\dfrac ab=n^2$ and $\dfrac aR=n$, then the correct equation(s) is(are)

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 charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)

A particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I ($a$ and $U_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

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 $T_M$. If the Young's modulus of the material of the wire is Y then $\dfrac1Y$ is equal to :

(g = gravitational acceleration)

For a simple pendulum, a graph is plotted between its kinetic energy (KE) and potential energy (PE) against its displacement d. Which one of the following represents these correctly ?

(graphs are schematic and not drawn to scale)

An LCR circuit is equivalent to a damped pendulum. In an LCR circuit the capacitor is charged to $Q_0$ and then connected to the L and R as shown below :

If a student plots graphs of the square of maximum charge ($Q^2_{\text{Max}}$) on the capacitor with time(t) for two different values $L_1$ and $L_2$ ($L_1>L_2$) of L then which of the following represents this graph correctly ? (plots are schematic and not drawn to scale)

$x$ and $y$ displacements of a particle are given as $x(t)=a\sin\omega t$ and $y(t)=a\sin2\omega t$. Its trajectory will look like :

A simple harmonic oscillator of angular frequency 2 rad s$^{-1}$ is acted upon by an external force $F=\sin t$ N. If the oscillator is at rest in its equilibrium position at $t=0$, its position at later times is proportional to :

A cylindrical block of wood (density = 650 kg m$^{-3}$), of base area 30 cm$^2$ and height 54 cm, floats in a liquid of density 900 kg m$^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

A pendulum with time period of 1s is losing energy due to damping. At certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s$^{-1}$) is :

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $K$ with time $t$ most appropriately? The figures are only illustrative and not to the scale.

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

Two bodies of masses 1 kg and 4 kg are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency 25 rad/s, and amplitude 1.6 cm while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take $g=10$ ms$^{-2}$).

The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to ($\ln5=1.601$, $\ln2=0.693$).

A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by ; $x=a_1\cos\omega t$ and $y=a_2\cos2\omega t$ traces a curve given by :

The angular frequency of the damped oscillator is given by,
$$\omega=\sqrt{\left(\frac km-\frac{r^2}{4m^2}\right)}$$
where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio $\dfrac{r^2}{mk}$ is 8%, the change in time period compared to the undamped oscillator is approximately as follows :

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $x$ is the displacement and a, b, c are positive constants ?