Oscillations practice · Mixed difficulty

The potential energy of a particle with displacement X is U(X). The motion is simple harmonic, when (K is a positive constant)

Two simple harmonic motions are represented by the equations
$x_1=5 \sin \left(2 \pi t+\frac{\pi}{4}\right), x_2=5 \sqrt{2}(\sin 2 \pi t+\cos 2 \pi t) .$
The ratio of amplitudes of $x_1$ and $x_2$ is

A particle performs simple harmonic motion with amplitude $A$. Its speed is tripled at the instant that it is at a distance $\frac{2}{3}A$ from equilibrium position. The new amplitude of the motion is

Sine and cosine functions are:

One end of a compressed spring of spring constant $5 \times 10^3 \mathrm{~N m}^{-1}$ is fixed to a wall and a block of mass $0.5 \mathrm{~kg}$ is kept in contact with the other end of the spring. If the spring is suddenly released, the block moves with a momentum $10 \mathrm{~Ns}$. Then the compression of the spring is

The ratio of K.E. of the particle executing S.H.M. at mean position to the K.E. at point whose distance is half of amplitude is:

The angular velocity and the amplitude of a simple pendulum is $\omega$ and $a$ respectively. At a displacement $x$ from the mean position, if its kinetic energy is $T$ and potential energy is $V$, then the ratio of $T$ to $V$ is

An experiment is performed to obtain the value of acceleration due to gravity $\mathrm{g}$ by using a simple pendulum of length $\mathrm{L}$. In this experiment time for 100 oscillations is measured by using a watch of 1 second least count and the value is 90.0 seconds. The length $L$ is measured by using a meter scale of least count 1 mm and the value is 20.0 cm. The error in the determination of $g$ would be:

The equation of a travelling wave is $y=15 \sin ^2(3 x-4 t),$ where $x, y$ are in meter and $t$ is in second. The amplitude of oscillation of particle is:

A body is moving in a room with a velocity of 20 m/s perpendicular to the two walls separated by 5 meters. There is no friction and the collisions with the walls are elastic. The motion of the body is

The length of the seconds pendulum is $1 \mathrm{~m}$ on earth. If the mass and diameter of the planet is 1.5 times that of the earth, the length of the seconds pendulum on the planet will be nearly.

The amplitude and the time period in a SHM is $0.5 \mathrm{~cm}$ and $0.4 \mathrm{~s}$, respectively. If the initial phase is $\pi / 2 \mathrm{~rad}$, then the equation of SHM will be

A child swinging on a swing in sitting position stands up, then the time period of the swing will

The displacement of an oscillator is given by 
$\mathrm{x}=\mathrm{a} \sin \omega \mathrm{t}+\mathrm{b} \cos \omega \mathrm{t}$
where $\mathrm{a}, \mathrm{b}$ and $\omega$, are constant. Then -

A mass m is suspended from a spring of length l and force constant K. The frequency of vibration of the mass is ${f_1}$. The spring is cut into two equal parts and the same mass is suspended from one of the parts. The new frequency of vibration of mass is ${f_2}$. Which of the following relations between the frequencies is correct

The total energy of a particle, executing simple harmonic motion is Where x is the displacement from the mean position.

A particle moves according to the law,$x = rcos\frac{{\pi t}}{{2.}}$ The distance covered by it the time interval between t= 0 to t=3s is

Assertion: Particle velocity and wave velocity both are independent of time.
Reason: For the propagation of wave motion, the medium must have the properties of elasticity and inertia.

A particle starts oscillating simple harmonically from its equilibrium position with time period $T$. The ratio of KE and PE of other particle at the t=$T$⁄12 is

A particle is subjected to two mutually perpendicular simple harmonic motions such that its $x$ and $y$ coordinates are given by $x=2 \sin \omega t,\,y=2 \sin \left(\omega t+\frac{\pi}{4}\right)$ The path of the particle will be

The path difference between the two waves: ${y_1} = {a_1}\,\sin \left( {\omega t - \frac{{2\pi x}}{\lambda }} \right)\,$ and ${y_2} = {a_2}\,\sin \left( {\omega t - \frac{{2\pi x}}{\lambda } + \phi } \right)\,$ will be

A pendulum of length L swings from rest to rest n times in one second. The value of acceleration due to gravity is

In a simple harmonic motion (SHM), which of
the following does not hold?

In case of simple harmonic motion, if the velocity is plotted along the $X$-axis and the displacement from the equilibrium position is plotted along the $Y$-axis, the resultant curve happens to be an ellipse with the ratio $\frac{\text { major axis }(\text { along } X)}{\text { minor axis }(\text { along } Y \text { ) }}=20 \pi$.
What is the frequency of the simple harmonic motion?

Total energy of a particle performing S.H.M. is 'NOT' proportional to

There is a body having mass $m$ and performing S.H.M. with amplitude $a$. There is a restoring force $F = - Kx$, where $x$ is the displacement. The total energy of body depends upon

The displacement $x$ of a body in motion is given by $x=a \sin (\omega t+\theta)$. The time at which the displacement is maximum is

The KE and PE of a particle executing SHM of amplitude $a$ will be equal when displacement is

During the phenomenon of resonance

Assertion: If a body is executing SHM, its acceleration is in the direction of velocity of body.
Reason: The velocity and acceleration of a body are acting in the same direction.