Oscillations practice · Mixed difficulty

Statement-I: The graph of potential energy and kinetic energy of a particle in SHM with respect to position is a parabola.

Statement-II: Potential energy and kinetic energy do not vary linearly with position.

Identify the function which represents a periodic motion.

The displacement of a particle is represented by the equation $y=\sin^3 \omega t$. The motion is

Assertion: If a pendulum falls freely, then its time period becomes infinite.
Reason: Free falling body has acceleration, equal to $g$.

The velocity of a particle performing simple
harmonic motion, when it passes through its
mean position is

A light spiral spring supports a 200 g weight at its lower end. It oscillates up and down with a period of 1 s. How much weight (in gram) must be removed from the lower end to reduce the period to 0.5 s?

Assertion (A): The bob of a simple pendulum is a ball full of water. If a fine hole is made at the bottom of the ball, then the time period will no more remain constant.
Reason (R): The time period of simple pendulum does not depend upon mass.

When an oscillator completes 100 oscillations its amplitude reduced to $\frac{1}{3}$ of the initial value. What will be its amplitude when it completes 200 oscillations?

In resonance:

If a body is performing SHM, then its

A) average kinetic energy per cycle is equal to half of its maximum kinetic energy

B) mean speed over a complete cycle is equal to $\frac{2}{\pi}$ times of its maximum speed

Assertion: All small oscillations are simple harmonic in nature.

Reason: Oscillations of spring block system are always simple harmonic whether amplitude is small or large.

Assertion: We can call circular motion also as simple harmonic motion.

Reason: Angular velocity in uniform circular motion and angular frequency in simple harmonic motion have the same meanings.

For definite length of wire, if the weight used for applying tension is immersed in water, then frequency will

The displacement of a particle executing periodic motion is given by $y=4 \cos ^2(t / 2) \sin (1000 t)$. Find the independent constituent SHM's.

Energy of particle executing SHM depends upon

A horizontal plank has a rectangular block placed on it. The plank starts oscillating vertically and simple harmonically with an amplitude of $40 {~cm}$. The block just loses contact with the plank when the later is momentarily at rest, then

The system that returns to equilibrium as quickly as possible without oscillating is

$x(t)=A \cos (\omega t+\phi)$ is the equation of simple harmonic motion. In this equation ' $\phi$ ' is called

S-I: If the amplitude of the oscillation is doubled the maximum kinetic energy becomes four times.
S-II: If the amplitude of the oscillation is doubled the minimum potential energy also becomes four times.

Instantaneous acceleration (in ${ms}^{-2}$ ) of a particle executing S.H.M is given by $a=-\frac{\pi}{4} \sin \left(\frac{\pi}{4} t-\frac{\pi}{6}\right)$. The maximum speed of the particle will occur first time at

The time period of oscillation of a simple pendulum of length $L$ suspended from the roof a vehicle, which moves without friction down an inclined plane of inclination $\alpha$, is given by:

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

Resonance is an example of:

A particle executes linear S.H.M. with amplitude $4 \mathrm{~cm}$. The magnitude of velocity and acceleration is equal when it is at $3 \mathrm{~cm}$ from mean position. The time period is

The period of a simple pendulum inside a stationary lift is$T$. The lift accelerates upwards with an acceleration of g⁄3. The time period of pendulum will be

Assertion (A): The graph of PE and KE of a particle is SHM with respect to position is a parabola.
Reason (R): This because PE and KE do not vary linearly with position.

The time period of a simple pendulum on a freely moving artificial satellite is

Which of the following quantities are always negative in a S.H.M
$(\overrightarrow F = \,force,\,\,\overrightarrow V \, = \,velocity,\,\,\overrightarrow a \, = \,acceleration,\,\,\overrightarrow r \, = \,displacement)$

Assertion: The graph of total energy of a particle in SHM. wrt, position is a straight line with zero slope.
Reason: Total energy of particle in SHM remains constant throughout its motion.

A particle of mass $4 {~cm}$ is executing oscillations about the origin on the $X$-axis. Its potential energy is $U(x)=k|x|^3$, where $k$ is a positive constant. If the amplitude of oscillation is $4 {~cm}$, then its time period $T$ will be proportional to