Oscillations (SHM) · JEE Main previous year questions

The displacement of a particle executing SHM is given by $x=10\sin \left(wt+\frac{\pi }{3}\right)m$. The time period of motion is $3.14\text{ s}$. The velocity of the particle at $t=0$ is ________ $\text{m/s}$.

If $\mathrm{R}$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $g={\pi }^{2}m/{\mathrm{s}}^2$, then the length of the second's pendulum at a height $\mathrm{h}=2R$ from the surface of earth will be :

1. A.$\frac{1}{9}\mathrm{m}$
2. B.$\frac{8}{9}\mathrm{m}$
3. C.$\frac{2}{9}\mathrm{m}$
4. D.$\frac{4}{9}\mathrm{m}$

A simple pendulum of length $1\mathrm{m}$ has a wooden bob of mass $1\mathrm{kg}$. It is struck by a bullet of mass $10^{-2}\mathrm{kg}$ moving with a speed of $2\times 10^2\mathrm{m}{\mathrm{s}}^{-1}$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $\mathrm{g}=10m/{\mathrm{s}}^2$ )

1. A.$0.20\mathrm{m}$
2. B.$0.40\mathrm{m}$
3. C.$0.30\mathrm{m}$
4. D.$0.35\mathrm{m}$

A mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9\text{ m}$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2}$ is ________ .

A particle performs simple harmonic motion with amplitude $A$. Its speed is increased to three times at an instant when its displacement is $\frac{2A}{3}$. The new amplitude of motion is $\frac{nA}{3}$. The value of $n$ is ____.

The measured value of the length of a simple pendulum is $20\mathrm{cm}$ with $2\mathrm{mm}$ accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is $\mathrm{N}\%$. The value of $\mathrm{N}$ is:

1. A.6
2. B.5
3. C.4
4. D.8

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}$, where $x=$ _______.

The bob of a pendulum was released from a horizontal position. The length of the pendulum is $10\text{ m}$. If it dissipates $10\%$ of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, $g:10{\text{ ms}}^{-2}$]

1. A.$5\sqrt{6}{\text{ ms}}^{-1}$
2. B.$5\sqrt{5}{\text{ ms}}^{-1}$
3. C.$2\sqrt{5}{\text{ ms}}^{-1}$
4. D.$6\sqrt{5}{\text{ ms}}^{-1}$

A simple harmonic oscillator has an amplitude $A$ and time period $6\pi \text{ second}$. Assuming the oscillation starts from its mean position, the time required by it to travel from $x=\mathrm{A}$ to $x=\frac{\sqrt{3}}{2}\text{ A}$ will be $\frac{\pi }{x}\text{ s}$, where $x=$ _________.

A particle executes simple harmonic motion with an amplitude of $4\text{ cm}$. At the mean position, velocity of the particle is $10\text{ cm/s}$. The distance of the particle from the mean position when its speed becomes $5\text{ cm/s}$ is $\sqrt{\alpha }\text{ cm}$, where $\alpha =$ _______.

At a given point of time the value of displacement of a simple harmonic oscillator is given as $y=\mathrm{A}\cos (30^{\circ })$. If amplitude is $40\text{ cm}$ and kinetic energy at that time is $200\text{ J}$, the value of force constant is $1.0\times 10^x{\text{ Nm}}^{-1}$. The value of $x$ is ___________.

A particle is executing simple harmonic motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be

1. A.1 : 1
2. B.1 : 4
3. C.2 : 1
4. D.1 : 3

1. A.
2. B.
3. C.
4. D.

The energy of an electromagnetic wave contained in a small volume oscillates with

1. A.double the frequency of the wave
2. B.the frequency of the wave
3. C.half the frequency of the wave
4. D.zero frequency

A rectangular block of mass 5 kg attached to a horizontal spiral spring executes simple harmonic motion of amplitude 1 m and time period 3.14 s. The maximum force exerted by spring on block is _________ N

1. A.$2\pi \sqrt{\frac{m}{K_1+K_2}}$
2. B.$2\pi \sqrt{\frac{m}{K_1-K_2}}$
3. C.$\frac{1}{2\pi }\sqrt{\frac{K_1+K_2}{m}}$
4. D.$\frac{1}{2\pi }\sqrt{\frac{K_1-K_2}{m}}$

A simple pendulum with length $100\text{ cm}$ and bob of mass $250\text{ g}$ is executing S.H.M. of amplitude $10\text{ cm}$. The maximum tension in the string is found to be $\frac{x}{40}\text{ N}$. The value of $x$ is ___________.

The maximum potential energy of a block executing simple harmonic motion is $25\text{ J}$. A is amplitude of oscillation. At $\mathrm{A}/2$, the kinetic energy of the block is

1. A.$9.75\text{ J}$
2. B.$37.5\text{ J}$
3. C.$18.75\text{ J}$
4. D.$12.5\text{ J}$

The general displacement of a simple harmonic oscillator is $x=A\sin \omega t$. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when $t=\frac{T}{\beta }$. The value of $\beta$ is __________.

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is _________ degree.

Identify the correct statements from the following : A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative. B. Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative. C. Work done by friction on a body sliding down an inclined plane is positive. D. Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero. E. Work done by the air resistance on an oscillating pendulum is negative. Choose the correct answer from the options given below :

1. A.A and C only
2. B.B and D only
3. C.B, D and E only
4. D.B and E only

A particle of mass 250 g executes a simple harmonic motion under a periodic force $\mathrm{F}=(-25x)\mathrm{N}$. The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ____ cm.

T is the time period of simple pendulum on the earth's surface. Its time period becomes $x$ T when taken to a height R (equal to earth's radius) above the earth's surface. Then, the value of $x$ will be :

1. A.4
2. B.$\frac{1}{2}$
3. C.2
4. D.$\frac{1}{4}$

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately) (Take $g=10\mathrm{m}{\mathrm{s}}^{-2}$, radius of earth = 6400 km)

1. A.12 hours
2. B.1 hour 24 minutes
3. C.24 hours
4. D.1 hour 40 minutes