Oscillations practice · Mixed difficulty

The potential energy of a particle (${U_x}$) executing SHM is given by

The length of a pendulum whose time period is 2 seconds at a place is increased by $3\,\%$ and it is taken to a place where acceleration due to gravity is $1\,\%$ less than the first place. Its new time period will be[in seconds]

The value of phase at maximum displacement from the mean position of a particle in S.H.M. is:

A spring produces extension ' $x$ ' by applying a force ' $F$ ' N. A body of mass ' $m$ ' suspended from spring oscillates vertically with a period ' $T$ ', the mass of the suspended body is (neglect mass of spring)

A ring of radius $20 \mathrm{~cm}$ is hinged from a point on its periphery. The time period of its oscillations will be $\left(g=9.8 \mathrm{~m} / \mathrm{s}^2\right)$

A lightly damped oscillator with a frequency $(\omega)$ is det in motion by harmonic driving force of frequency $(n)$. When $n < \omega$, then response of the oscillator is controlled by

A particle executes simple harmonic motion with a frequency $f$ The frequency with which its kinetic energy oscillates is

Maximum speed of a particle in simple harmonic motion is $v_{\max }.$ Then average speed of a particle in SHM for complete cycle is equal to

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

A particle of mass $m$ moves in a force field such that its potential energy in force field is defined by the equation $U$ $=+\mathrm{A}(\mathrm{x}-\mathrm{a})^2(\mathrm{x}-\mathrm{b})^2$. Where $\mathrm{A}, \mathrm{a}$ and $\mathrm{b}$ are $+\mathrm{ve}$ constants then body may oscillate simple harmonically about point.

The equation $y=A \sin ^2(k x-\omega t)$ represents a wave with:

A particle is executing SHM of period 24 $x$ and of amplitude 41 cm with O as equilibrium position. The minimum time in seconds taken by the particle to go from P to Q. where OP = -9 cm and OQ =40 cm is

A simple pendulum consisting of a ball of mass $m$ tied to a string of length $l$ is made to swing on a circular arc of angle $\theta$ in a vertical plane. At the end of this arc, another ball of mass $m$ is placed at rest. The momentum translated to this ball at rest by the swinging ball is

A pendulum is oscillating at a frequency of 8 Hz. Suddenly the string of the pendulum is clamped at its midpoint then the new frequency of oscillation is.

Two simple harmonic motions are represented by ${y_1} = 6\sin \left( {2\pi t + \frac{\pi }{3}} \right)$ and ${y_2} = 3(\sin 2\pi t + \cos 2\pi t).$ The ratio of their amplitudes is

The percentage error in the measurement of ${g}$ is:

(Given that ${g}=\frac{4 \pi^2 {L}}{{T}^2}, {~L}=(10 \pm 0.1) \mathrm{cm}, \, {T}=(100 \pm 1) \mathrm{s})$

If the length of the Equator is about $40000 \mathrm{~km}$ and the velocity of rotation is about $1700 \mathrm{~km}$ per hour, what would be the velocity of rotation at the pole?

The periods of oscillations of a pendulum at two places are 1.4 S and 1.6 S. The ratio of two places are 1.4 S and 1.6 S. The ratio of the accelerations due to gravity at those two places

Assertion(A): If amplitude of SHM is increased, time period of SHM will increase.

Reason(R): If amplitude is increased, body will have to travel more distance in one complete oscillation.

A block of mass $M$ is suspended from a light spring of force constant $k$. Another mass $m$ moving upwards with velocity $v$ hits the mass $M$ and gets embedded in it. What will be the amplitude of oscillation of the combined mass?

A simple pendulum is oscillating with amplitude $A$ and angular frequency $\omega$. At displacement $x$ from mean position, the ratio of kinetic energy to potential energy is:

Time-period of a pendulum on a satellite, orbiting around the earth, is

Assertion (A): In SHM, acceleration is always directed towards the mean position.
Reason (R): The body stops momentarily at the extreme positions and then moves back to mean position.

If the differential equation given by $\frac{{{d^2}y}}{{d{t^2}}} + 2k\frac{{dy}}{{dt}} + {\omega ^2}y = {F_0}\sin pt$ Describes the oscillatory motion of body in a dissipative medium under the influence of a periodic force, then the state of maximum amplitude of the oscillation is a measure of

Under the influence of a periodic force $\mathrm{F} 1$ a particle performs $\mathrm{SHM}$ along $\mathrm{x}$-axis and its $\mathrm{x}$ coordinate varies with time as, $x=A \sin \omega t$. When the force $F 1$ is replaced by another periodic force F2, particle performs SHM along $y$-axis and its $y$-coordinate varies with time as, $y=B \sin (\omega t+\Phi)$. When both forces are applied on the particle simultaneously, the particle performs SHM along a straight line in the $x-y$ plane.

(i) The value of $\Phi$ may be zero

(ii) The value of $\Phi$ may be $\pi$

(iii) The slope of the resulting path may be $\frac{B}{A}$

(iv) The slope of the resulting path may be $-\frac{B}{A}$

Choose the correct option.

A particle is moving with SHM in a straight line. When the distance of the particle from the equilibrium position has values $x_1$ and $x_2$, the corresponding values of velocities are $u_1$ and $u_2$. the time period of oscillation $T=$

A body performing simple harmonic motion has potential energy ' $P_1$ ' at displacement ' $x_1$ '. Its potential energy is ' ${P}_{{2}}$ ' at displacement ' ${x}_{{2}}$ '. The potential energy ' $P$ ' at displacement $\left({x}_1+{x}_2\right)$ is

A simple pendulum, suspended from the ceiling of a stationary van, has time period T. If the van starts moving with a uniform velocity the period of the pendulum will be

The total energy of a particle executing S.H.M. is proportional to

The distance travelled by a particle in one oscillation, executing SHM with amplitude $\mathrm{A}$, is