Chapter Questions
Any position of sine antable coefficients. Which of the periods time penous Whaticians proved this result? following(b) Carl Friedrich Gauss pythagoras(a) Pythag (c) Leonhard Euler fean Baptiste Joseph Fourier(d) Jean bapat
Halley's comet appears after every The(a) 72 years(b) 74 years(c) 76 years(d) 78 years
(c) Which one of the the earth about its axis.(a) Rotation(b) A freely suspended bar magnet displaced from and released. its $\mathrm{N}-\mathrm{S}$ direction of hands of a clock.(c) Motion of hanus of from a bow.(d) An arrow
Which of the following $x$-t graphs does not represent periodic motion?
If the frequency of human heart is $1.25 \mathrm{~Hz}$, the number of heart beats in 1 minute is(a) 65(b) 75(c) 80(d) 90
In a certain oscillatory system, the amplitude of motion is $5 \mathrm{~m}$ and the time period is $4 \mathrm{~s}$. The time taken by the particle for passing between points which are at distances of $4 \mathrm{~m}$ and $2 \mathrm{~m}$ from the centre and on the same side of it will be(a) $0.30 \mathrm{~s}$(b) $0.32 \mathrm{~s}$(c) $0.33 \mathrm{~s}$(d) $0.35 \mathrm{~s}$
A block of mass $m$ is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is $4 \mathrm{~cm}$ below the point, where it was held in hand. The amplitude of the oscillation is(a) $1 \mathrm{~cm}$(b) $2 \mathrm{~cm}$(c) $3 \mathrm{~cm}$(d) $4 \mathrm{~cm}$
Out of the following functions representing motion of a particle which represents SHM?1. $x=\sin ^{3} \omega t$2. $x=1+\omega t+\omega^{2} t^{2}$3. $x=\cos \omega t+\cos 3 \omega t+\cos 5 \omega t$4. $x=\sin \omega t+\cos \omega t$(a) Only 1(b) Only 1 and 3(c) Only 1 and 4(d) Only 4
Which of the following is not a characteristics of simple harmonic motion?(a) The motion is periodic.(b) The motion is along a straight line about the mean position.(c) The oscillations are responsible for the energy conversion.(d) The acceleration of the particle is directed towards the extreme position.
The equation of motion of a simple hamonik motion is(a) $\frac{d^{2} x}{d t^{2}}=-c^{2} x$(b) $\frac{d^{2} x}{d t^{2}}=-v^{2} t$(c) $\frac{d^{2} x}{d t^{2}}=-c u x$(d) $\frac{d^{2} x}{d t^{2}}=-\mathrm{cot}$
Which of the following expressions represent simple harmoni motion?(c) $x=B \sin (\omega t+\beta)$(d) $x=$ Asincut cos'ot
If a simple harmonic motion is represented by $\frac{d^{2} x}{d t^{2}}+c x=0$, its time period is(a) $2 \pi \sqrt{c}$(b) $2 \pi c$(c) $\frac{2 \pi}{\sqrt{\alpha}}$(d) $\frac{2 \pi}{\alpha}$
The time period of simple harmonic motion dependsupon(a) amplitude (b) encist(c) phase constant(d) $\mathrm{mass}$
Which of the following motions is not simple harmonic?(a) Vertical oscillations of a spring(b) Motion of a simple pendulum(c) Motion of planet around the sun(d) Oscillation of tiquid in a U-tube
If $x, v$ and a represent the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $T$ then which of the following does not change?(a) $\frac{a T}{v}$(b) $a T+2 \pi v$(c) $\frac{a T}{x}$(d) $a^{2} T^{2}+4 \pi^{2} v^{2}$
The function sincot - costat represents(a) a simple harmonic motion with a period $\frac{\pi}{\omega}$.(b) a simple harmonic motion with a period $\frac{2 \pi}{\omega}$.(c) a periodic, but not simple harmonic motion with a period $\frac{\pi}{\omega}$,(d) a periodic, but not simple harmonic motion with a period $\frac{2 \pi}{\omega}$.
A particle executing simple harmonic motion with an amplitude $A$. The distance travelled by the particle in one time period is(a) zero(b) $A$(c) $2 A$(d) $4 A$
Displacement vs time curve for a particle sin shown in the figure.which of the following statements is correc??(a) Phase of the oscillator is same at $t=0 \mathrm{~s}$ and
Two simple harmonic motions are represented the equations. $y_{1}=10 \sin \frac{\pi}{4}(12 t+1), y_{2}=5\left(\sin 3 \mathrm{p} t+\sqrt{3} \cos \frac{3}{\mathrm{p}}\right)$The ratio of their amplitudes is(a) $1: 1$(b) $1: 2$(c) $3: 2$(d) $2: 3$
Figure shows the displacement-time graphs of the simple harmonic motions I and II. From the graph it follows that(a) curve I has same frequency as that of curvell.(b) curve I has frequency twice that of curve ll.(c) curve I has frequency half that of curve II.(d) curve I has frequency four times that of curvell.
A vibratory motion is represented by$$\begin{aligned}x=2 A \cos () t+A \cos \left(\omega t+\frac{\pi}{2}\right)+& A \cos (\omega t+\pi) \\&+\frac{A}{2} \cos \left(\omega t+\frac{3 \pi}{2}\right) .\end{aligned}$$The resultant amplitude of the motion is(a) $\frac{9 \mathrm{~A}}{2}$(b) $\frac{\sqrt{5} A}{2}$(c) $\frac{5 A}{2}$(d) $2 \mathrm{~A}$
A particle executing SHM is described by the displacement function $x(t)=A \cos (\omega t+\phi)$ If the initial $(t=0)$ position of the particle is $I \mathrm{~cm}$, its initial velocity is $\pi \mathrm{cm} \mathrm{s}^{-1}$ and its angular frequency is $\pi \mathrm{s}^{-1}$, then the amplitude of its motion is(a) $\pi \mathrm{cm}$(b) $2 \mathrm{~cm}$(c) $\sqrt{2} \mathrm{~cm}$(d) $1 \mathrm{~cm}$
Two particles execute simple harmonic motions of same amplitude and frequency along the same straight line. They cross one another when goingThe phase difference between Whectionts. displacements are one half of their osite ain theirof $\mathrm{f}^{0}$ when(ci) $150^{\circ}$is $\quad$ (c) $120^{\circ}$the plitudes is (b) $30^{\circ}$ $60^{\circ}$ ne $\mathrm{SHM}$ of same amplitude and
$60^{2}$ sute $\mathrm{SHM}$ of same amplitude and(b) $\frac{2 \pi}{3}$(a) $2 \mathrm{~kg}$ is attached to the spring $o$
$2 \mathrm{~kg}$ is attached to the sprmz= block of mass $20 \mathrm{~N} \mathrm{~m}^{-1}$. The block is pulled to a spring constant son of $5 \mathrm{~cm}$ from its equilibrium position at distance of 2 , on a horizontal frictionless surface from rest at $x=0$ on The displacement of the block at any time t is $t=0$(a) $x=0.05 \sin 5 t \mathrm{~m}$(b) $x=0.05 \cos 5 t \mathrm{~m}$$x=0.5 \sin 5 t \mathrm{~m}$(d) $x=5 \sin 5 t \mathrm{~m}$(c) $x^{2}=0$ is filled in a tube
(c) $x=0.2$ liquid of density $\rho$ is filled in a tube viscous liquio cross section, as shown in the A non viscer areaIf the liquid is slightly depressed in one of the with $A$ as oscillates with a frequency figure. the liquid column arms.
The circular motion of a particle with constant speedis(a) periodic and simple harmonic.(b) simple harmonic but not periodic.(c) neither periodic nor simple harmonic.(d) periodic but not simple harmonic.
Simple harmonic motion is the projection of uniform circular motion on(a) $x$-axis(b) $y$-axis(c) reference circle(d) any diameter of reference circle
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated in the figure. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle $P$ is(a) $x=2 \cos \left(2 \pi t+\frac{\pi}{4}\right)$(b) $x=2 \sin \left(2 \pi t+\frac{\pi}{4}\right)$(c) $x=2 \sin \left(2 \pi t-\frac{\pi}{4}\right)$(d) $x=2 \cos \left(2 \pi t-\frac{\pi}{4}\right)$
In an SHM, $x$ is the displacement and $a$ is the acceleration at time $t$. The plot of $a$ against $x$ for one complete oscillation will be(a) a straight line(b) a circle(c) an ellipse(d) a sinusoidal curve
Which one of the following statements is true for the velocity $v$ and the acceleration $a$ of a particle executing simple harmonic motion?(a) When $v$ is maximum, $a$ is zero.(b) When $v$ is zero, $a$ is zero.(c) When $v$ is maximum, $a$ is maximum.(d) Value of $a$ is zero, whatever may be the value of $v$.
Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle executing simple harmonic motion?(a) $a=2 x^{2}$(b) $a=-2 x^{2}$(c) $a=2 x$(d) $a=-2 x$
A particle executing simple harmonic motion wion an amplitude $A$ and angular frequency $\omega$. The ratio of maximum acceleration to the maximum velocity of the particle is(a) $\omega A$(b) $\omega^{2} \mathrm{~A}$(c) $\omega$(d) $\frac{\omega^{2}}{A}$
The displacement-time graph for a particle executing SHM is as shown in figure.Which of the following statements is correct?(a) The velocity of the particle is maximum at $t=\frac{3}{4} T$(b) The velocity of the partide is maximum at $t=\frac{T}{2}$(c) The acceleration of the particle is maximum at $t=\frac{T}{4}$(d) The acceleration of the particle is maximum at $t=\frac{3}{4} T$
Displacement versus time curve for a particle executing SHM is as shown in figure.At what points the velocity of the particle is zero?(a) $A, C, E$(b) $B, D, F$(c) $A, D, F$(d) $C, E, F$
A particle executing SHM with time period $T$ and amplitude $A$. The mean velocity of the particle averaged over quarter oscillation is(a) $\frac{A}{4 T}$(b) $\frac{2 A}{T}$(c) $\frac{3 A}{T}$(d) $\frac{4 A}{T}$
A particle is in linear simple harmonic motion $A O=O B=5 \mathrm{~cm}$between two points $A$ and $B C=8 \mathrm{~cm}$$B, 10 \mathrm{~cm}$ apart (figure). Take the direction from $A$ to $B$ as the tve direction. Which of the following statements is correct?(a) The sign of acceleration and force on the particle when it is at $A$ is negative.(b) The sign of acceleration and force on the particle when it is at $B$ is positive.(c) The sign of velocity, acceleration and force on the partide when it is $3 \mathrm{~cm}$ away from $A$ going towards $B$ are positive.
A particle executing SHM. The phase 56between velocity and displacement is(a) 0(b) $\frac{\pi}{2}$(c) $\pi$(d) 2
A particle executing SHM. The phase difet between acceleration and displacement is(a) 0(b) $\frac{\pi}{2}$(c) $\pi$(d) $\frac{3}{2^{7}}$
A block of mass $m$ is attached to a spring of constant $k$ is free to oscillate with angular of $\mathrm{spn}$ in a horizontal plane without friction or clachy, It is pulled to a distance $x_{0}$ and pushed toward centre with a velocity $v_{0}$ at time $t=0$. The anplitud of oscillations in terms of $\omega, x_{0}$ and $v_{0}$ is(a) $\sqrt{\frac{v_{0}^{2}}{\omega^{2}}-x_{0}^{2}}$(b) $\sqrt{\omega^{2} v_{0}^{2}+x_{0}^{2}}$(c) $\sqrt{\frac{x_{0}^{2}}{\omega^{2}}+v_{0}^{2}}$(d) $\sqrt{\frac{v_{0}^{2}}{\omega^{2}}+x_{0}^{2}}$
The piston in the cylinder head of a locomotiz has a stroke of $6 \mathrm{~m}$. If the piston executing simple harmonic motion with an angular frequency of $200 \mathrm{rad} \mathrm{min}^{-1}$, its maximum speed is(a) $5 \mathrm{~m} \mathrm{~s}^{-1}$(b) $10 \mathrm{~m} \mathrm{~s}^{-1}$ (c) $15 \mathrm{~m} \mathrm{~s}^{-1}$(d) $20 \mathrm{~m} \mathrm{~s}^{-1}$
A particle executing SHM according to the equation $x=5 \cos \left(2 \pi t+\frac{\pi}{4}\right)$ in SI units. The displacemet and acceleration of the particle at $t=1.5 \mathrm{~s}$ is(a) $-3.0 \mathrm{~m}, 100 \mathrm{~m} \mathrm{~s}^{-2}$(b) $+2.54 \mathrm{~m}, 200 \mathrm{~m} 5^{2}$(c) $-3.54 \mathrm{~m}, 140 \mathrm{~m} \mathrm{~s}^{-2}$(d) $+3.55 \mathrm{~m}, 120 \mathrm{~m} \mathrm{~s}^{-2}$
A mass oscillates along the $x$-axis according to the law, $x=x_{0} \cos \left(\omega t-\frac{\pi}{4}\right)$. If the acceleration of the particle is written as $a=A \cos (\omega t+\delta)$, then(a) $A=x_{0} \omega^{2}, \delta=\frac{3 \pi}{4}$(b) $A=x_{0}, \delta=-\frac{\pi}{4}$(c) $A=x_{0} \omega^{2}, \delta=\frac{\pi}{4}$(d) $A=x_{0} \omega^{2}, \delta=-\frac{\pi}{4}$
The $x-t$ graph of a particle undergoing simple harmonic motion is as shown in the figure.of the particle at $t=\frac{4}{3} \mathrm{~s}$ is acceleration The accer(a) $\frac{\sqrt{3}}{32} \pi^{2} \mathrm{~cm} \mathrm{~s}^{-2}$(d) $-\frac{\sqrt{3}}{32} \pi^{2} \mathrm{~cm} \mathrm{~s}^{-2}$(c) $\frac{\pi^{2}}{32} \mathrm{~cm} \mathrm{~s}^{-2}$
Displacement of a particle executing simple motion is given by$x$ is in metres where $x$ is and maximum speed of the particle is
(c) 5 executes SHM of period 12 s. After two A particle executes of oscillation, geconds, it passes. thevelocity is found to be $3.142 \mathrm{~cm} \mathrm{~s}^{-1}$. The amplitudeof oscillations is(c) $24 \mathrm{~cm}$(d) $12 \mathrm{~cm}$(b) $3 \mathrm{~cm}$(a) $6 \mathrm{~cm}$ (D) 2
(2) executing simple harmonic motion withA particle amplitude $5 \mathrm{~cm}$ and a time period $0.2 \mathrm{~s}$. The an and acceleration of the particle when thevelocity and acco$\begin{array}{ll}\text { displacement is } 5 \mathrm{~cm} \text { is } \\ \text { (a) } 0.5 \pi \mathrm{m} \mathrm{s}^{-1}, 0 \mathrm{~m} \mathrm{~s}^{-2} & \text { (b) } 0.5 \mathrm{~m} \mathrm{~s}^{-1},-5 \pi^{2} \mathrm{~m} \mathrm{~s}^{-2}\end{array}$(c) $0 \mathrm{~m} \mathrm{~s}^{-1},-5 \pi^{2} \mathrm{~m} \mathrm{~s}^{-2}$(d) $0.5 \pi \mathrm{m} \mathrm{s}^{-1},-0.5 \pi^{2} \mathrm{~m} \mathrm{~s}^{-2}$
of mass $3 \mathrm{~kg}$ is under tension of $400 \mathrm{~N}$. The length of the stretched string is $25 \mathrm{~cm}$. If the transverse jerk is stuck at one end of the string how longdoes the disturbance take to reach the other end?(a) $0.047 \mathrm{~s}$(b) $0.055 \mathrm{~s}$(c) $0.034 \mathrm{~s}$(d) $0.065 \mathrm{~s}$
9. Natural length of the spring is $40 \mathrm{~cm}$ and its spring constant is $4000 \mathrm{~N} \mathrm{~m}^{-1}$. A mass of $20 \mathrm{~kg}$ is hung from it. The extension produced in the spring is (Given $\left.g=9.8 \mathrm{~m} \mathrm{~s}^{-2}\right)$(a) $4.9 \mathrm{~cm}$(b) $0.49 \mathrm{~cm}$(c) $9.4 \mathrm{~cm}$(d) $0.94 \mathrm{~cm}$
In simple harmonic motion, at the extreme positions(a) kinetic energy is minimum, potential energy is maximum.(b) kinetic energy is maximum, potential energy is minimum.(c) both kinetic and potential energies are maximum.(d) both kinetic and potential energies are minimum.
The total energy of a simple harmonic oscillator is Proportional to(a) amplitude(b) square of amplitude(c) frequency(d) velocity
A particle of mass $m$ executing SHM with amplitude $A$ and angular frequency $\omega$. The average value of the kinetic energy and potential energy over a period is(a) $0, \frac{1}{2} m u^{2} A^{2}$(b) $\frac{1}{2} m u^{2} A^{2}$,(c) $\frac{1}{2} m \omega^{2} A^{2}, \frac{1}{2} m \omega^{2} A^{2}$(d) $\frac{1}{4} m \omega^{2} A^{2}, \frac{1}{4} m x^{2} A^{2}$
A simple harmonic oscillator has a period $T$ and energy $E$. The amplitude of the oscillator is doubled. Choose the correct answer.(a) Period and energy get doubled.(b) Period gets doubled while energy remains the same.(c) Energy gets doubled while period remains the same.(d) Period remains the same and energy becomes four times.
A particle executing simple harmonic motion with time period $T$. The time period with which its kinetic energy oscillates is(a) $\mathrm{T}$(b) $2 \mathrm{~T}$(c) $4 T$(d) $\frac{T}{2}$
A particle executing SHM with an amplitude $A$. The displacement of the particle when its potential energy is half of its total energy is(a) $\frac{A}{\sqrt{2}}$(b) $\frac{A}{2}$(c) $\frac{\mathrm{A}}{4}$(d) $\frac{A}{3}$
A block of mass $m$ is hanging vertically by spring of spring constant $k$. If the mass is made to oscillate vertically, its total energy is(a) maximum at the extreme position(b) maximum at the mean position(c) minimum at the mean position(d) same at all positions
Frequency of variation of kinetic energy of a simple harmonic motion of frequency $n$ is(a) $2 n$(b) $n$(c) $\frac{n}{2}$(d) $3 n$
For a particle executing simple harmonic motion, the displacement $x$ is given by $x=A \cos \omega t$. Identify the graph, which represents the variation of potential energy (U) as a function of time t and displacement $x$.(a) I, III(b) II, III(c) I, IV(d) II,IV
When the displacement of a particle executing SHM is one-fourth of its amplitude, what fraction of the total energy is the kinetic energy?(a) $\frac{16}{15}$(b) $\frac{15}{16}$(c) $\frac{3}{4}$(d) $\frac{4}{3}$
A block whose mass is $1 \mathrm{~kg}$ is fastened to a spring. The spring has a spring constant of $100 \mathrm{~N} \mathrm{~m}^{-1}$. 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 $t=0 .$ The kinetic energy and potential energy of the block when it is $5 \mathrm{~cm}$ away from the mean position is(a) $0.0375 \mathrm{~J}, 0.125 \mathrm{~J}$(b) $0.125 \mathrm{~J}, 0.375 \mathrm{~J}$(c) $0.125 \mathrm{~J}, 0.125 \mathrm{~J}$(d) $0.375 \mathrm{~J}, 0.375 \mathrm{~J}$
A body of mass $m$ is situated in a potential field $U(x)=U_{0}(1-\cos \alpha x)$ where $U_{0}$ and $\alpha$ are constants. The time period of small oscillations is(a) $2 \pi \sqrt{\frac{m}{U_{0} \alpha}}$(b) $2 \pi \sqrt{\frac{m}{U_{0} \alpha^{2}}}$(c) $2 \pi \sqrt{\frac{m}{2 U_{0} \alpha}}$(d) $2 \pi \sqrt{\frac{2 m}{U_{0} \alpha^{2}}}$
The frequency of oscillations of a mass $m$ suspended by a spring is $v_{1}$. If the length of the spring is cut to one-half, the same mass oscillates with frequency $v_{2} .$ The value $v_{2} / v_{1}$ is(a) 2(b) $\sqrt{2}$(c) 4(d) $\sqrt{3}$
Time period of oscillation of a spring is $12 \mathrm{~s}$ on earth. What shall be the time period if it is taken to moon?(a) $6 \mathrm{~s}$(b) $12 \mathrm{~s}$(c) $36 \mathrm{~s}$(d) $72 \mathrm{~s}$
A $5 \mathrm{~kg}$ collar is attached to a spring of spring constant $500 \mathrm{~N} \mathrm{~m}^{-1}$. It slides without friction over a horizontal rod. The collar is displaced from its equilibrium position by $10 \mathrm{~cm}$ and released. The time period of oscillation is
In the question number 64 , 231acceleration of the collar is oq, the (a) $5 \mathrm{~m} \mathrm{~s}^{-2}$(c) $15 \mathrm{~m} \mathrm{~s}^{-2}$(b) $10 \mathrm{~m} \mathrm{~s}^{-2}$(d) $20 \mathrm{~m} \mathrm{~s}^{-2}$
A body of mass $20 \mathrm{~g}$ connected to a spring of constant $k$, executes simple harmonic mo of frequency of $(5 / \pi) \mathrm{Hz}$. The value of(a) $4 \mathrm{~N} \mathrm{~m}^{-1}$ (b) $3 \mathrm{~N} \mathrm{~m}^{-1}$(c) $2 \mathrm{~N}_{\mathrm{m}-1}$Two blocks each of mass $m_{j}$ (d) $5 \mathrm{~N}$
Two blocks each of mass $m$ is connected spring of spring constant $k$ as shown in the foIf the blocks are displaced slightly in directions and released, they will execute harmonic motion. The time period of(a) $2 \pi \sqrt{\frac{m}{k}}$(b) $2 \pi \sqrt{\frac{m}{2 k}}$
An air chamber of volume $V$ has a neck of crost sectional area $a$ into which a light ball of $\mathrm{mass}$ just fits and can move up and down without friction The diameter of the ball is equal to that of the ned of the chamber. The ball is pressed down a little and released. If the bulk modulus of air is $B$, the time period of the oscillation of the ball is(a) $T=2 \pi \sqrt{\frac{B a^{2}}{m V}}$(b) $T=2 \pi \sqrt{\frac{B V}{m a^{2}}}$(c) $T=2 \pi \sqrt{\frac{m B}{V a^{2}}}$(d) $T=2 \pi \sqrt{\frac{m V}{B a^{2}}}$
A spring balance has a scale that reads from 0 to $50 \mathrm{~kg}$. The length of the scale is $20 \mathrm{~cm}$. A block of mass $m$ is suspended from this balance, when displaced and released, it oscillates with a period $0.5 \mathrm{~s}$. The value of $m$ is (Take $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ )(a) $8 \mathrm{~kg}$(b) $12 \mathrm{~kg}$(c) $16 \mathrm{~kg}$(d) $20 \mathrm{~kg}$
A trolley of mass $3 \mathrm{~kg}$, as shown in figure, is connected to two identical springs, each of spring constant $600 \mathrm{~N} \mathrm{~m}^{-1}$. If the trolley is displaced from its equilibrium position by $5 \mathrm{~cm}$ and released, the maximum speed of the trolley is(a) $0.5 \mathrm{~m} \mathrm{~s}^{-1}$(b) $1 \mathrm{~m} \mathrm{~s}^{-1}$(c) $2 \mathrm{~m} \mathrm{~s}^{-1}$(d) $3 \mathrm{~m} \mathrm{~s}^{-1}$
A tray on two racmed on of spring shown in figure. springs. constant $k$, as tray is depressed a When the tray ised, it executes and released, motion of period $1.5 \mathrm{~s}$. When a little simple harmon $m$ is placed on the tray, the period of block os cillation becomes $3 \mathrm{~s}$. The value of $m$ is(b) $20 \mathrm{~kg}$$\begin{array}{ll}10 \mathrm{~kg} & \text { (d) } 40 \mathrm{~kg}\end{array}$(a) $30 \mathrm{~kg}$(c) $30 \mathrm{AB}$
Two identical springs of spring constant $k$ areTo block of mass $m$ and to fixed supports attached to a block in the figure. The time period of oscillation as hownis$\pi$(a) $2 \pi \sqrt{\frac{m}{k}}$(b) $2 \pi \sqrt{\frac{m}{2 k}}$(c) $2 \pi \sqrt{\frac{2 m}{k}}$(d) $\pi \sqrt{\frac{m}{2 k}}$
Match the Column I with Column II. Column II(a) $A-p, B-q, C-s, D-r$(b) $A-s, B-r, C-p, D-q$(c) $\mathrm{A}-\mathrm{r}, \mathrm{B}-\mathrm{p}, \mathrm{C}-\mathrm{s}, \mathrm{D}-\mathrm{q}$(d) $A-p, B-r, C-q, D-s$
The time period of mass M when displaced from Inextensible its equilibrium position string and then released for the system as shown in figure is(a) $2 \pi \sqrt{\frac{M}{k}}$(b) $2 \pi \sqrt{\frac{M}{2 k}}$(c) $2 \pi \sqrt{\frac{M}{4 k}}$(d) $2 \pi \sqrt{\frac{2 M}{k}}$
To show that a simple pendulum executing a simple harmonic motion it is necessary to assume that(a) length of the pendulum is small.(b) mass of the pendulum is small.(c) acceleration due to gravity is small.(d) amplitude of the oscillation is small.
Motion of an oscillating liquid in a $U$ tube is(a) periodic but not simple harmonic.(b) non-periodic.(c) simple harmonic and time period is independent of the density of the liquid.(d) simple harmonic and time period is directly proportional to the density of the liquid.
What is the effect on the time period of a simple pendulum if the mass of the bob is doubled?(a) Halved(b) Doubled(c) Becomes 8 times(d) No effect
The acceleration due to gravity on the surface of the moon is $1.7 \mathrm{~m} \mathrm{~s}^{-2}$. The time period of a simple pendulum on the moon if its time period on the earth is $3.5 \mathrm{~s}$ is (Given, $g=9.8 \mathrm{~m} \mathrm{~s}^{-2}$ ).(a) $2.2 \mathrm{~s}$(b) $4.4 \mathrm{~s}$(c) $8.4 \mathrm{~s}$(d) $16.8 \mathrm{~s}$
Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme position making an angle of $2^{\circ}$ to the right with the vertical, the other pendulum makes an angle of $1^{\circ}$ to the left of the vertical. The phase difference between the pendulums is(a) $\frac{\pi}{2}$(b) $\frac{2}{3} \pi$(c) $\frac{3}{2} \pi$(d) $\pi$
Two pendulums differ in lengths by $22 \mathrm{~cm}$. They oscillate at the same place so that one of them makes 30 oscillations and the other makes 36 oscillations during the same time. The lengths (in $\mathrm{cm}$ ) of thependulums are$$\begin{array}{ll}\text { (b) } 60 \text { and } 38\end{array}$$(a) 72 and 50(c) 50 and 28(d) 80 and 58
The time period of a simple pendulum on the surface of the earth is $4 \mathrm{~s}$. Its time period on the surface of the moon is(a) $4 \mathrm{~s}$(b) $s s$(c) $10 \mathrm{~s}$(d) $12 \mathrm{~s}$
A disc of radius $R=10 \mathrm{~cm}$ oscillates as a physical pendulum about an axis perpendicular to the plane of the disc at a distance $r$ from its centre. If $r=\frac{R}{4}$, the approximate period of oscillation is (Take $\left.g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$(a) $0.84 \mathrm{~s}$(b) $0.94 \mathrm{~s}$(c) $1.26 \mathrm{~s}$(d) $1.42 \mathrm{~s}$
A simple pendulum suspended from the roof of a lift oscillates with frequency $v$ when the lift is at rest. If the lift talls freely under gravity, its frequency of oscillation becomes(a) zero(b) $v$(c) 20(d) infinite
A simple pendulum of length $L$ and having a bob of mass $m$ is suspended in a car. The car is moving on a circular track of radius $R$ with a uniform speed $v$. If the pendulum makes small oscillations in a radial direction about its equilibrium position, its time period of oscillation is(a) $T=2 \pi \sqrt{\frac{L}{g}}$(b) $T=2 \pi \sqrt{\frac{L}{\sqrt{g^{2}+\frac{v^{4}}{R^{2}}}}}$(c) $T=2 \pi \sqrt{\frac{L}{\sqrt{g^{2}+\frac{v^{2}}{R}}}}$(d) $T=2 \pi \sqrt{\frac{L}{g^{2}-\frac{v^{4}}{R^{2}}}}$
A simple pendulum executing SHM with a period of $6 \mathrm{~s}$ between two extreme positions $B$ and $C$ about a point $O$. If the length of the arc $B C$ is $10 \mathrm{~cm}$, how long will the pendulum take the move from position $C$ to a position $D$ towards $O$ exactly midway between $C$ and $O ?$(a) $0.5 \mathrm{~s}$(b) $1 \mathrm{~s}$(c) $1.5 \mathrm{~s}$(d) $3 \mathrm{~s}$
The length of a seconds pendulum on the surface of earth is $1 \mathrm{~m}$. Its length on the surface of the moon is(a) $\frac{1}{6} \mathrm{~m}$(b) $1 \mathrm{~m}$(c) $\frac{1}{36}$ r(d) $36 \mathrm{~m}$
The length of the simple pendulum which ticks seconds is(a) $0.5 \mathrm{~m}$(b) $1 \mathrm{~m}$(c) $1.5 \mathrm{~m}$(d) $2 \mathrm{~m}$
A rectangular block of mass $m$ and area of crosssection $A$ floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period $T$. Then(a) $T \propto \frac{1}{\sqrt{m}}$(b) $T_{\infty} \sqrt{p}$(c) $T \propto \frac{1}{\sqrt{A}}$(d) $T \propto \frac{1}{\rho}$
A sphere of mass $m$ makes SHM in a hemispherical bowl $A B C$ and it moves from $A$ to $C$ and back to $A$ via $A B C$, so that $P B=h$. If acceleration due to gravity is $g$ the speed of when it just crosses the point $B$ is(a) $2 g h$(b) $m g h$(c) $\sqrt{2 \mathrm{gh}}$(d) $g h$
A particle oscillating under a force $\vec{F}=-k \vec{x}-b_{i}$ is $(k$ and $b$ are constants)(a) simple harmonic oscillator(b) linear oscillator(c) damped oscillator(d) forced oscillator
Which of the following displacement-time graphs represent damped harmonic oscillation?
Which of the following energy-time graphs represents damped harmonic oscillator?
of mass $200 \mathrm{~g}$ executing SHM under thefor its amplitude to drop to half of its initial value is $\ln (1 / 2)=-0.693)$$\begin{array}{llll}\text { (Given, In (lil) } & \text { (b) } 9 \mathrm{~s} & \text { (c) } 4 \mathrm{~s} & \text { (d) } 11 \mathrm{~s}\end{array}$(a) $7 \mathrm{~s}$
In the question number 93 , the time elapsed for its mechanical energy to drop half of its initial value is(a) $2.5 \mathrm{~s}$(b) $3.5 \mathrm{~s}$(c) $4.5 \mathrm{~s}$(d) $7.5 \mathrm{~s}$
In case of force oscillations of a body force is constant throughout.(a) driving(b) driving force is to be applied only momentarily.(c) driving force has to be periodic and continuous.(d) driving force is not required.
Resonance is an example of(a) forced oscillation(b) damped oscillation(c) free oscillation(d) none of these In case
In case of a forced oscillation, the resonance peak becomes very sharp when the(a) restoring force is small.(b) damping force is small.(c) quality factor is small.(d) applied periodic force is small.
At resonance, the amplitude of forced oscillations is(a) minimum(b) maximum(c) zero(d) none of these
Which of the following statements is correct?(a) Every periodic motion is simple harmonic motion.(b) In simple harmonic motion the period is proportional to the square of the amplitude of oscillation.(c) In simple harmonic motion the phase constant depends on initial condition.(d) The resonance frequency of a driven oscillator depends on the damping.
A block of mass $m$ is attached to a spring of spring constant $k$ and has a natural frequency $\omega_{0}$. An external force $F(t)$ proportional to cos wt $\left(\omega \neq \omega_{0}\right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to(a) $\frac{m}{\omega_{0}^{2}-\omega^{2}}$(b) $\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)}$(c) $\frac{1}{m\left(\omega_{0}^{2}+\omega^{2}\right)}$(d) $\frac{m}{\omega_{0}^{2}+\omega^{2}}$