Simple Harmonic Oscillation

Q. A system exhibiting S.H.M. must have

1. elasticity as well as inertia
2. elasticity, inertia and an external force
3. elasticity only
4. inertia only

Q. A particle is executing linear SHM of amplitude $A$ and time period $T$. If $v$ refers to the average speed of the particle during any time interval of $\frac{T}{3}$, then the maximum possible value of $v$ in terms of $A$ is

1. $\frac{3\sqrt{3}}{T}A$
2. $\frac{\sqrt{3}A}{T}$
3. $\frac{2\sqrt{3}}{T}A$
4. $\frac{3A}{T}$

Q. Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle implies simple harmonic motion?

1. $a=0.7x$
2. $a=-200x^2$
3. $a=-10x$
4. $a=100x^3$

Q. The potential energy of a particle of mass $0.1\text{kg}$, moving along the $x-$axis, is given by $U=5x(x-4)\text{J}$, where $x$ is in meters.
Choose the wrong option(s).

1. The speed of the particle is maximum at $x=2\text{m}$.
2. The particle executes simple harmonic motion.
3. The period of oscillation of the particle is $\frac{\pi }{5}s$.
4. The particle does not execute simple harmonic motion.

Q. The equation of a particle executing SHM is $y=20\sin (2\pi t-\frac{\pi }{3})$.
The amplitude $A$, time period $T$ and angular frequency $\omega$ are
[All quantities are in SI units]

1. $A=10\text{m},T=1\text{sec},\omega =2\pi \text{rad/sec}$
2. $A=20\text{m},T=2\text{sec},\omega =2\pi \text{rad/sec}$
3. $A=40\text{m},T=1\text{sec},\omega =2\pi \text{rad/sec}$
4. $A=20\text{m},T=1\text{sec},\omega =2\pi \text{rad/sec}$

Q. A particle is subjected to two simple harmonic motion along $x$ and $y$ directions, according to equation $x=3\sin 100\pi t,y=4\sin 100\pi t$. Then,

1. motion of particle will be on ellipse travelling in clockwise direction.
2. Motion of particle will be on a straight line with slope $\frac{4}{3}$
3. Motion will be a simple harmonic motion with amplitude $5$.
4. Phase difference between two motions is $\frac{\pi }{2}$.

Q. A circular spring of natural length $l$ is cut and welded with two beads of masses $m_1$ and $m_2$ such that the ratio of the lengths of the springs between the beads is $4:1$. If the stiffness of the original spring is $k$, then find the angular frequency of oscillation of the beads in a smooth horizontal rigid tube.
[Assume $m_1=m$ and $m_2=3m$].

1. $\omega =\sqrt{\frac{25k}{3m}}$
2. $\omega =\sqrt{\frac{5k}{3m}}$
3. $\omega =\sqrt{\frac{25k}{4m}}$
4. $\omega =\sqrt{\frac{5k}{4m}}$

Q. How does the speed and magnitude of acceleration change, when a particle performing simple harmonic motion moves from mean position to one of its extreme positions ?

1. Speed increases, magnitude of acceleration decreases
2. Speed decreases, magnitude of acceleration increases
3. Speed increases, magnitude of acceleration increases
4. Speed decreases, magnitude of acceleration decreases

Q. A street car moves rectilinearly from station A (here car stops) to the next station B (here also car stops ) with an acceleration varying according to the law $f=a-bx$, where $a$ and $b$ are positive constants and $x$ is the distance from station A. The distance between the two stations & the maximum velocity are respectively.

1. $x=\frac{2a}{b};v_{max}=\frac{a}{\sqrt{b}}$
2. $x=\frac{a}{b};v_{max}=\frac{a}{2\sqrt{b}}$
3. $x=\frac{2a}{b};v_{max}=\frac{2a}{\sqrt{b}}$
4. $x=\frac{a}{2b};v_{max}=\frac{a}{\sqrt{b}}$

Q. A block with a mass of $3\text{kg}$ is suspended from an ideal spring having negligible mass which stretches the spring by $0.2\text{m}$ at equilibrium. Find the force constant of the spring.

1. $147\text{N/m}$
2. $247\text{N/m}$
3. $347\text{N/m}$
4. $47\text{N/m}$

Q. A dipole consists of two particles one with charge $+1\mu C$ and mass $1\text{kg}$ and the other with charge $-1\mu C$ and mass $2\text{kg}$ separated by a distance of 3 m. For small oscillations about its equilibrium position, the angular frequency, when placed in a uniform electric field of $20\text{kV/m}$ is

1. $0.1\text{rad/s}$
2. $1.1\text{rad/s}$
3. $10\text{rad/s}$
4. $2.5\text{rad/s}$

Q. A particle is executing linear SHM of amplitude $A$ and time period $T$. If $v$ refers to the average speed of the particle during any time interval of $\frac{T}{3}$, then the maximum possible value of $v$ in terms of $A$ is

1. $\frac{3\sqrt{3}}{T}A$
2. $\frac{\sqrt{3}A}{T}$
3. $\frac{2\sqrt{3}}{T}A$
4. $\frac{3A}{T}$

Q. A spring has natural length of $50\text{cm}$ and a force constant of $2.0\times {10}^3{\text{Nm}}^{-1}$ . A body of mass $10\text{kg}$ is suspended from it in equilibrium position. If the body is pulled down, such that length of spring becomes $58\text{cm}$ and released. If it executes simple harmonic motion, what is the net force on the body, when it is at it's lowermost position of oscillation ?
$(\text{Take g}=10{\text{ms}}^{-2})$.

1. $20\text{N}$
2. $40\text{N}$
3. $60\text{N}$
4. $80\text{N}$

Q. A particle is subjected to two simple harmonic motion along $x$ and $y$ directions, according to equation $x=3\sin 100\pi t,y=4\sin 100\pi t$. Then,

1. motion of particle will be on ellipse travelling in clockwise direction.
2. Motion of particle will be on a straight line with slope $\frac{4}{3}$
3. Motion will be a simple harmonic motion with amplitude $5$.
4. Phase difference between two motions is $\frac{\pi }{2}$.

Q. A cylindrical piston of mass $M$ slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas.
The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

1. $T=2\pi \sqrt{\frac{Mh}{PA}}$
2. $T=2\pi \sqrt{\frac{MA}{Ph}}$
3. $T=2\pi \sqrt{\frac{M}{PAh}}$
4. $T=2\pi \sqrt{MPHA}$

Q. Out of the following functions representing motion of a particle which represent S.H.M?
$I.y=sin\omega t-cos\omega t$
$II.y=si{n}^3\omega t$
$III.y=5cos(\frac{3\pi }{4}-3\omega t)$
$IV.y=1+{\omega }^2{t}^2+\omega t$

1. Only $\left(IV\right)$ does not represent S.H.M.
2. $\left(I\right)$ and $\left(III\right)$
3. $\left(I\right)$ and $\left(II\right)$
4. Only $\left(I\right)$

Q. A small planet is between two stars of different masses and radii such that the net force on the planet is zero. On displacing the planet on either side, can it under go oscillatory motion?

1. No, not at all
2. Only if the displacement is small
3. Yes definitely
4. One can't surely say

Q. The speed $v$ of a particle moving along a straight line, when it is at a distance $x$ from a fixed point on the line, is given by $v^2=144-9x^2$.
Choose the wrong option.

1. $\text{Displacement of the particle}\le \text{distance moved by it}$
2. $\text{The magnitude of acceleration at a distance 3 units from the fixed point is 27 units}$
3. $\text{The motion is simple harmonic with}T=\frac{\pi }{3}\text{units}$
4. $\text{The maximum displacement from the fixed point is 4 units}$

Q. A small planet is between two stars of different masses and radii such that the net force on the planet is zero. On displacing the planet on either side, can it under go oscillatory motion?

1. No, not at all
2. Only if the displacement is small
3. Yes definitely
4. One can't surely say

Q. A force $F=(a+2x-bx)\text{N}$ is acting on a body of mass $4\text{kg},$ which is moving in a straight line. Is the body moving in simple harmonic motion ? If yes, find the equilibrium position of the particle.
[Assume $a>2,b>2$]

1. No
2. $\text{Yes},\frac{a}{(b+2)}$
3. $\text{Yes},\frac{(b-2)}{a}$
4. $\text{Yes},\frac{a}{(b-2)}$

Q. The speed $v$ of a particle moving along a straight line, when it is at a distance $x$ from a fixed point on the line, is given by $v^2=144-9x^2$.
Choose the wrong option.

1. $\text{Displacement of the particle}\le \text{distance moved by it}$
2. $\text{The magnitude of acceleration at a distance 3 units from the fixed point is 27 units}$
3. $\text{The motion is simple harmonic with}T=\frac{\pi }{3}\text{units}$
4. $\text{The maximum displacement from the fixed point is 4 units}$

Q. Statement 1: $v=u+at$ can be used as an equation for SHM.
Statement 2: Acceleration in SHM is variable.

1. Statement 1 is true, statement 2 is false
2. Statement 1 is false, statement 2 is true
3. Statement 1 and statement 2 are true
4. Statement 1 and statement 2 are false

Q.
A ball is rolling on a mountain of the shape as shown in the picture. At which place will the ant be in stable equilibrium?

1. A
2. A and D
3. C
4. E

Q. A small block of mass $m$ is fixed at upper end of a massless vertical spring of spring constant $k=\frac{4mg}{L}$ and natural length ${}^{\prime }10L^{\prime }$. The lower end of spring is free and is at a height $L$ from the fixed horizontal floor as shown in figure. The spring is initially unstretched and the spring-block system is released from rest in the shown position. At the instant speed of block is maximum, the magnitude of force exerted by spring on the block is

1. $\frac{mg}{2}$
2. $mg$
3. zero
4. None of these

Q. In the figure shown, find the maximum amplitude of the system, so that there is no slipping between any of the blocks.

1. $\frac{2}{7}\text{m}$
2. $\frac{3}{4}\text{m}$
3. $\frac{4}{9}\text{m}$
4. $\frac{10}{3}\text{m}$

Q. A spring block system is kept on a frictionless surface. Match the respective entries of column (I) with column (II) assuming minimum potential energy for the system as zero.
($K$ = Spring constant, $A$ = Amplitude, $m$ = Mass of block)
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) If mass of the block is doubled}& \text{(p) Time period increases}\\ \text{(keeping}K\text{,}A\text{unchanged)}& \\ \text{(B) If the amplitude of oscillation is doubled}& \text{(q) Time period decreases}\\ \text{(keeping}K\text{,}m\text{unchanged)}& \\ \text{(C) If force constant is doubled}& \text{(r) Energy of oscillation increases}\\ \text{(keeping}m\text{,}A\text{unchanged)}& \\ \text{(D) If another spring of same force constant}& \text{(s) Energy of oscillation decreases}\\ \text{is attached parallel to the previous one}& \\ \text{(keeping}m\text{,}A\text{unchanged)}& \text{(t) Energy of oscillation remain constant}\end{array}$

1. $\left(A\right)\to (r,t);\left(B\right)\to (r,t);\left(C\right)\to (p,q);\left(D\right)\to (q,t)$
2. $\left(A\right)\to \left(s\right);\left(B\right)\to (p,q);\left(C\right)\to (q,r);\left(D\right)\to (p,t)$
3. $\left(A\right)\to (p,t);\left(B\right)\to \left(r\right);\left(C\right)\to (q,r);\left(D\right)\to (q,r)$
4. $\left(A\right)\to (q,t);\left(B\right)\to \left(s\right);\left(C\right)\to (p,q);\left(D\right)\to (q,r)$

Q. In SHM, the acceleration of the object must be directly proportional to its displacement from some fixed point.

1. False
2. True