Oscillations practice · Medium

A block of mass $1 \mathrm{~kg}$ is fastened to a spring of spring constant of $100 \mathrm{~Nm}^{-1}$. The block is pulled to a distance $x=10 \mathrm{~cm}$ from its equilibrium position $(x=0 \mathrm{~cm})$ on a frictionless surface from rest $t=0$. The kinetic energy and the potential energy of the block when it is $5 \mathrm{~cm}$ away from the mean position is

The displacement of a particle (in meter) from its mean position is given by the equation $y=0.2\left(\cos ^2 \frac{\pi t}{2}-\sin ^2 \frac{\pi t}{2}\right)$. The motion of the above particle is

A pendulum having time period equal to two seconds is called a seconds pendulum. Those used in pendulum clocks are of this type. Find the length of a seconds pendulum at a place where $g=\pi^2 \mathrm{~m} \mathrm{~s}^{-2}$.

Identify the wrong statement from the following?

Average velocity of a particle executing SHM in one complete vibration is:

When a particle executes SHM, there is always a constant ratio between its displacement and

Statement I The total energy of a particle executing simple harmonic motion, can be negative.

Statement II Potential energy of a system can be negative.

Assertion: In a SHM, kinetic and potential energies become equal when the displacement is $1 / \sqrt{2}$ times the amplitude.
Reason: In SHM, kinetic energy is zero when potential energy is maximum.

Statement-I: During the oscillation of a simple pendulum the tension in string is greatest at the mean position.
Statement-II: At the mean position the centripetal acceleration of the bob is maximum.

The instantaneous displacement of a simple pendulum oscillator is given by $x=A \cos \left(\omega t+\frac{\pi}{4}\right)$. Its speed will be maximum at time :-

A particle executes simple harmonic motion between $x$ $= - A\,\,\mathrm{and}\,\,x = + A$ The time taken for it to go from $0\mathrm{ }to\;A/2\,\,is\,\,{T_1}$ and to go from $A/2$ to $A\,\,\mathrm{is}\,\,{T_2}$ Then

Assertion (A): The time period of a simple pendulum of infinite length is finite value.
Reason (R): The time period of a simple pendulum is directly proportional to the square root of length.

Motion of an oscillating liquid column in a $\mathrm{U}$-tube is

Which of the following quantity does NOT
change due to damping of oscillations?

A particle executing simple harmonic motion of amplitude 5 cm has maximum speed of 31.4 cm/s. The frequency of the oscillation is

Starting from mean position a body oscillates simple harmonically with a period of $6 \mathrm{~s}$. After what time, the kinetic energy will be one third of potential energy?

At what displacement from mean position is the K.E. of a particle performing SHM of amplitude $30 \mathrm{~cm}$, eight times its P.E.

A seconds pendulum has a hollow spherical bob of mass $25 \times {10^{ - 3}}kg.$ It is replaced by another small solid bob of same radius but of mass $50 \times {10^{ - 3}}kg$. The new time period will be.

Two spring of spring constant $k_1$ and $k_2$ are joined in series. The effective spring constant of the combination is given by

The amplitude of a damped oscillator varies with time as $A(t)={A}_0 \exp (-b t / 2 m)$ where $b=$ $70 \mathrm{~g} / \mathrm{s}$ and $m=200 \mathrm{~g}$. How long does it take for the mechanical energy to drop to one - fourth of its initial value? [Take ln 2 = 0.7]

A pendulum has time period $T$ in air. When
it is made to oscillate in water, it acquired
a time period $T' = \sqrt 2 T$. The density of the
material pendulum bob is equal to

A particle moving along the $x$ axis executes simple harmonic motion, then the force acting on it is given by

A pendulum is swinging in an elevator. Its
period will be greatest when the elevator is

A body having natural frequency $\nu$' is executing forced oscillations under a driving force of frequency. The system will vibrate

If a body is executing simple harmonic motion then

The function ( $\sin \omega t-\cos \omega t)$ represents the S.H.M having a time period $T$

A particle moving about its equilibrium position with equation $y=-a x-b t$. Interpret the condition

The acceleration of particle in SHM:

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