Oscillations practice · Mixed difficulty

Particle moves on the $\mathrm{x}$-axis according to the equation $\mathrm{x}=$ $\mathrm{A}+\mathrm{B} \sin \omega \mathrm{t}$. The motion is simple harmonic with amplitude.

How much revolution does the engine make during the time when a motor wheel with angular speed is increased from $720 \mathrm{~rpm}$ to $2820 \mathrm{~rpm}$ in 14 seconds?

An object of mass $0.2 \mathrm{~kg}$ executes $\mathrm{SHM}$ along $x$-axis with frequency of $\frac{25}{\pi}$ Hz. At the position $x=0.04 \mathrm{~m}$, the object has kinetic energy of $0.5 \mathrm{~J}$ and potential energy of $0.4 \mathrm{~J}$. The amplitude of oscillation is

The amplitude of vibration of a particle is given by ${a_m} = ({a_0})/(a{\omega ^2} - b\omega + c);$ where ${a_0},a,b$ and c are positive. The condition for a single resonant frequency is

The height of liquid column in a $U$ tube is 0.3 meter. If the liquid in one of the limbs is depressed and then released, then the time period of liquid column will be-

If the ratio of the acceleration due to gravity on the surface of earth to that on the surface of the moon is $6: 1$, then the ratio of the periods of a simple pendulum on their surface is

If two springs $A$ and $B$ with spring constant $2 \mathrm{~k}$ and $\mathrm{k}$, are stretched separately by same suspended weight, then the ratio between the work done in stretching $A$ and $B$ is

The phase difference between two $SHMs$ $y_1=10 \sin \left(20 t+\frac{\pi}{6}\right)$ and $y_2=20 \cos \left(20 t-\frac{\pi}{6}\right)$ is

A horizontal platform vibrates with simple harmonic motion in the horizontal direction with a period 2 s. A body of mass 0.5 kg is placed on the platform. The coefficient of static friction between the body and platform is 0.3. What is the maximum frictional force on the body when the platform is oscillating with amplitude 0.2 m? Assume ${\pi ^2} = 10 = g$.

A ceiling fan rotates about its own axis with some angular velocity. When the fan is switched off, the angular velocity becomes $\left(\frac{1}{4}\right)^{\text {th }}$ of the original in time ' $t$ ' and ' $n$ ' revolutions are made in that time. The number of revolutions made by the fan during the time interval between switch off and rest are (Angular retardation is uniform)

A body is executing simple harmonic motion. When the displacements from the mean position is $4 {~cm}$ and $5 {~cm}$, the corresponding velocities of the body are $10 {~cm} / {sec}$ and $8 {~cm} / {sec}$. Then the time period of the body is

The total mechanical energy of a spring-mass system in simple harmonic motion is $E=\frac{1}{2} m \omega^2 A^2$. Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude $A$ remains the same. The new mechanical energy will

A heavy sphere of mass $m$ is suspended by string of length $l$. The sphere is made to revolve about a vertical line passing through the point of suspension in a horizontal circle such that the string always remains inclined to the vertical at an angle $\theta$. What is its period of revolution?

The acceleration of a particle in S.H.M. is

A block of mass one $\mathrm{kg}$ is fastened to a spring with a spring constant $50 \mathrm{~Nm}^{-1}$. The block is pulled to a distance $x=10 \mathrm{~cm}$ from its equilibrium position at $x=0$ on a frictionless surface from rest at $t=0$. Write the expression for its $x(t)$ and $v(t)$.

Which one of the following is not a periodic motion?

A particle starts S.H.M. from the mean position. Its amplitude is a and total energy ${E}$. At one instant 5 its kinetic energy is $3 {~E} / 4$, its displacement at this instant is

Two particles are executing SHMs. The equations of their motions are

$y_1=10 \sin \left(\omega t+\frac{\pi}{4}\right) \text { and } y_2=5 \sin \left(\omega t+\frac{\sqrt{3} \pi}{4}\right)$

What is the ratio of their amplitudes?

Assertion: Every periodic motion is not always a simple harmonic motion.
Reason: The motion governed by the force law $F = −kx$ is simple harmonic.

If a simple pendulum executing SHM falls freely along with the support, then

The acceleration of particle executing SHM
when it is at mean position is

Two particles execute simple harmonic motion of the same amplitude and frequency along the same straight line. They pass one another when going in opposite directions and each time their displacement is half the amplitude. The phase difference between them is

A particle oscillates simple harmonically with a period of $16 \mathrm{~s}$. Two second after crossing the equilibrium position its velocity becomes $1 \mathrm{~m} / \mathrm{s}$. The amplitude is

The equation of a simple harmonic motion is $x=0.34 \cos (3000 t+0.74)$, where $x$ and $t$ are in $\mathrm{mm}$ and second, respectively. The frequency of the motion (in $\mathrm{Hz}$) is

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

S-I: In SHM, total mechanical energy can be negative also.
S-II: Potential energy is always negative.

The equation of a simple harmonic motion is given by $y=6 \sin 10 \pi t+8 \cos 10 \pi t$, where $y$ is in cm and $t$ in sec. Determine the period and initial phase.

A particle executing simple harmonic motion has an amplitude of $6 \mathrm{~cm}$. Its acceleration at a distance of $2 \mathrm{~cm}$ from the mean position is $8 \mathrm{cms}^{-2}$. The maximum speed of the particle is

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?

Statement-I: In simple harmonic motion $A$ is the amplitude of oscillation. If $t_1$ be the time to reach the particle from mean position to $\frac{A}{\sqrt{2}}$ and $t_2$ the time to reach from $\frac{A}{\sqrt{2}}$ to $A$. Then $t_1=\frac{t_2}{\sqrt{2}}$

Statement-II: Equation of motion for the particle starting from mean position is given by $x= \pm \mathrm{A} \sin \omega \mathrm{t}$ and of the particle starting from extreme position is given by $x= \pm$ $A\cos \omega t$