Time Period Independent of Amplitude

Q. A ring of radius $\left(R\right)$ carries a uniformly distributed charge $+Q$. A point charge $-q$ is placed on the axis of the ring at a distance $2R$ from the centre of the ring and released from rest. The particle

1. stays in rest
2. executes simple harmonic motion
3. executes oscillation but not SHM
4. moves to the centre of ring immediately and stays there

Q. The $x-t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=\frac{4}{3}\text{s}$ is

1. $\frac{\sqrt{3}{\pi }^2}{32}{\text{cm/s}}^2$
2. $-\frac{{\pi }^2}{32}{\text{cm/s}}^2$
3. $\frac{{\pi }^2}{32}{\text{cm/s}}^2$
4. $-\frac{\sqrt{3}{\pi }^2}{32}{\text{cm/s}}^2$

Q. A particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta .$ Its time period of vibration will be

1. $\frac{2\pi \beta }{\alpha }$
2. $\frac{{\beta }^2}{{\alpha }^2}$
3. $\frac{\alpha }{\beta }$
4. $\frac{{\beta }^2}{\alpha }$

Q. The potential energy of a particle of mass $1\text{kg}$ in motion along the $x-$ axis is given by $U=4(1-\text{cos}2x)\text{J}$, where $x$ is in $\text{m}$. The period of small oscillations (in $\text{s}$) is

1. $2\pi$
2. $3\pi$
3. $\pi$
4. $\frac{\pi }{2}$

Q. A simple pendulum is executing simple harmonic motion with a time period T. If the length of the pendulum is increased by $21\%,$ the percentage increase in the time period of the pendulum of increased length is

1. 21%
2. 10%
3. 30%
4. 50%

Q. The equation of motion of a particle executing simple harmonic motion is $a+16{\pi }^{2}x=0$. In this equation, $a$ is linear acceleration in ${\text{m/s}}^2$ of the particle at displacement $x$ in $\text{m}$. The time period of this simple harmonic motion is

1. $\frac{1}{4}\text{s}$
2. $\frac{1}{2}\text{s}$
3. $\frac{1}{3}\text{s}$
4. $\frac{1}{5}\text{s}$

Q. A particle executes SHM with a time period of $4\text{s}$. Find the time taken by the particle to go directly from its mean position to half of its amplitude.

1. $\frac{1}{6}\text{s}$
2. $\frac{1}{3}\text{s}$
3. $\frac{1}{2}\text{s}$
4. $\frac{2}{5}\text{s}$

Q. A neutron, a proton, an electron and an $\alpha$ particle enter a region of constant magnetic field with equal velocity. The $\overrightarrow{B}$ is along the perpendicularly inward to the plane of the paper. The path followed by each particle is shown in the figure.


The path followed by $\alpha$ particle is represented by:

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

Q. The time period of polar satellites is about (take $g=9.8{\text{ms}}^{-2}$ and radius of the Earth
$R=6.3\times {10}^6\text{m}$),

[Height of a polar satellite above earth's surface will be around $700\text{km}$]

1. $24\text{hr}$
2. $1000\text{min}$
3. $100\text{min}$
4. $6\text{hr}$

Q. A particle performing SHM with amplitude $A$ and time period $T$ starts from mean position towards the positive extreme point as shown in the figure. Find the time taken by the particle (in seconds) to reach $x=\frac{A}{2}.$

1. $t=\frac{T}{6}$
2. $t=\frac{T}{8}$
3. $t=\frac{T}{4}$
4. $t=\frac{T}{12}$

Q. A particle is executing SHM with time period $T$, starting from mean position. Time taken by it to complete $\frac{5}{8}\text{th}$ part of oscillation will be

1. $\frac{T}{12}$
2. $\frac{T}{6}$
3. $\frac{5T}{12}$
4. $\frac{7T}{12}$

Q. A horizontal platform with an object placed on it is executing S.H.M. in the vertical direction. The amplitude of oscillation is $3.92\times {10}^{-3}\text{m}.$ What should be the least period of these oscillations, so that the object is not detached from the platform ?

1. $0.1256\text{sec}$
2. $0.1456\text{sec}$
3. $0.1556\text{sec}$
4. $0.1356\text{sec}$

Q.

A large horizontal surface moves up and down in SHM with an amplitude of 1 cm. If a mass of 10 kg (which is placed on the surface) is to remain continually in contact with it, the maximum frequency of S.H.M. will be

1. 0.5 Hz

2. 5 Hz

3. 1.5 Hz

4. 10 Hz

Q. 1. The motion of a particle executing simple harmonic motion is described by the displacement function, $x\left(t\right)=A\cos (\omega t+\varphi )$. If the initial $(t=0)$ position of the particle is $1cm$ and its initial velocity is $\omega cm/s$, what are its amplitude and initial phase angle ? The angular frequency of the particle is $\pi s^{-1}$.


2. If instead of the cosine function, we choose the sine function to describe the SHM : $x=B\sin (\omega t+\alpha )$, what are the amplitude and initial phase of the particle with the above initial conditions.

Q. A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2$. The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the figure. If the block is given a displacement downwards and released, then neglecting friction, which of the following statement(s) is/are correct?

1. Block executes simple harmonic motion.
2. The block's motion is periodic but not simple harmonic.
3. The frequency of oscillation is independent of the size of the block.
4. The motion of the block is symmetric about its equilibrium position.

Q. A point charge $-q$ having mass $m$ is placed at the centre of a uniformly charged ring of charge $Q$ and radius $R$. Find the time period of oscillation if the point charge is slightly displaced and released.

1. $T=2\pi \sqrt{\frac{m{R}^3}{KQq}}$
2. $T=2\pi \sqrt{\frac{m{R}^2}{KQq}}$
3. $T=2\pi \sqrt{\frac{m{R}^3}{2KQq}}$
4. None of the above

Q. A particle is performing S.H.M with time period $T$. Find the shortest time taken by the particle to go from the mean position to an extreme position.

1. $\frac{T}{4}$
2. $\frac{T}{8}$
3. $\frac{T}{2}$
4. $\frac{T}{6}$

Q.

A simple pendulum is made of a body which is a hollow sphere containing mercury suspended by means of a wire. If a little mercury is drained off, the period of pendulum will

]

1. Remains unchanged

2. Increase

3. Decrease

4. Become erratic

Q. The value of 120 joule per second on a system that has 10 g , 100cm , 1000 second as base unit is

Q. A ring of mass $M$, radius $R$, cross-section area $‘a^{\prime }$ and Young's modulus $Y$ is kept on a smooth cone of base radius $2R$ and semi vertex angle ${45}^{\circ }$, as shown in figure. Assume that the extension in the ring is small. Pick the correct alternative $\left(s\right)$.

1. The tension in the ring will be same throughout.
2. The tension in the ring will be independent of its radius.
3. The elongation in the ring is $\frac{MgR}{aY}$
4. The elastic potential energy stored in the ring will be $\frac{M^2{g}^{2}R}{8\pi Ya}$

Q. The maximum velocity and acceleration of a particle in $\text{SHM}$ are $100\text{cm/s}$ and $157{\text{cm/s}}^2$ respectively. The time period in seconds will be :

1. $4$
2. $1.57$
3. $0.25$
4. $1$

Q.

A rod of length L and mass M is acted on by two unequal forces $F_1$ and $F_2(<F_1)$ as shown in the figure. The tension in the rod at a distance y from the end A is given by:

1. $F_1[1-\frac{y}{L}]+F_2\left[\frac{y}{L}\right]$

2. $(F_1-F_2)\frac{y}{L}$

3. $F_2[1-\frac{y}{L}]+F_1\left[\frac{y}{L}\right]$

4. None of these

Q. A particle is executing SHM along a straight line. Its velocities at distances $x_1$ and $x_2$ from the mean position are $v_1$ and $v_2$ respectively. Its time period of oscillation is

1. $2\pi \sqrt{\frac{x_2^2-x_1^2}{v_1^2-v_2^2}}$
2. $2\pi \sqrt{\frac{v_1^2+v_2^2}{x_1^2+x_2^2}}$
3. $2\pi \sqrt{\frac{v_1^2-v_2^2}{x_1^2-x_2^2}}$
4. $2\pi \sqrt{\frac{x_1^2+x_2^2}{v_1^2+v_2^2}}$

Q.

The function of time representing a simple harmonic motion with a period of $\frac{\mathrm{\pi }}{\omega }$ is

1. $\cos \left(\omega t\right)+\cos \left(2\omega t\right)+\cos \left(3\omega t\right)$

2. $3\cos \left(\frac{\mathrm{\pi }}{4}-2\omega t\right)$

3. ${\sin }^2\omega t$

4. $\sin \left(\omega t\right)+\cos \left(\omega t\right)$

Q. A simple harmonic motion of amplitude A has a time period T. The acceleration of the oscillator when its displacement is half the amplitude is

1. 
2. 
3. -
4. -

Q. The time period of oscillation positively a charged spherical bob of a simple pendulum in a downward uniform electric field is.

1. $T=2\pi \sqrt{\frac{l}{g}}$
2. $T=2\pi \sqrt{\frac{l}{(g+\frac{Eq}{m})}}$
3. $T=2\pi \sqrt{\frac{l}{(g-\frac{Eq}{m})}}$
4. $T=2\pi \sqrt{\frac{l}{mg+Eq}}$

Q. A brass cube of side $a$ and density $\sigma$ is floating in mercury of density $\rho$. If the cube is slightly displaced vertically, it executes $SHM$. Its time period will be

1. $2\pi \sqrt{\frac{\sigma a}{\rho g}}$
2. $2\pi \sqrt{\frac{\rho a}{\sigma g}}$
3. $2\pi \sqrt{\frac{\rho g}{\sigma a}}$
4. $2\pi \sqrt{\frac{\sigma g}{\rho a}}$

Q. A block of mass $m$ compresses a spring of stiffness constant $k$ and natural length $l$, through a distance $\frac{l}{2}$ as shown in the figure. If the block is not fixed with the spring, the period of motion of the block is
[Assume collision to be elastic with the wall and horizontal surface to be smooth]

1. $(\pi +4)\sqrt{\frac{m}{K}}$
2. $2\pi \sqrt{\frac{m}{K}}$
3. None of these
4. $(1+\pi )\sqrt{\frac{m}{k}}$

Q.

An open organ pipe has a length of 5 cm. (a) Find the fundamental frequency of vibration of this pipe. (b) What is the highest harmonic of such a tube that is in the audible range ? Speed of sound in air is $340m{s}^{-1}$ and the audible range is 20-20, 000 Hz.

Q.

A balloon going upwards with a velocity of 12m/s is at ht.65 m from earth at any instant.At this instant a packet drops from it.how much time will packet take in reaching earth?