CBSE Questions for Class 11 Engineering Physics Oscillations Quiz 13 - MCQExams.com

A block of mass $$m$$ has initial velocity $$u$$ having direction towards '+x axis'. The block stops after covering a distance $$S$$ causing similar extension in the spring of constant $$K$$ holding it, $$\mu$$ is the kinetic friction between the block and the surface on which it was moving, the distance $$S$$ is given by: 
984472_ec0d4c74cd8540fc8307704a46e57a66.png
  • $$\dfrac{1}{K}\mu^2m^2g^2$$
  • $$\dfrac{1}{K}(mKu^2 - \mu^2m^2g^2)^{1/2}$$
  • $$\dfrac{1}{K}(\mu^2m^2g^2 + mK\mu^2 + \mu mg)^{1/2}$$
  • $$\dfrac{\mu mg + \sqrt{\mu^2m^2g^2 + mu^2k}}{k}$$
An air chamber of volume V has a neck of cross-sectional area a into which a light ball of mass m just fits and can move up and down without friction. The diameter of the ball is equal to that of the neck 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
  • $$T = 2 \pi \sqrt{\frac{Ba^2}{mV}}$$
  • $$T = 2 \pi \sqrt{\frac{BV}{ma^2}}$$
  • $$T = 2 \pi \sqrt{\frac{mB}{Va^2}}$$
  • $$T = 2 \pi \sqrt{\frac{mV}{Ba^2}}$$

Two blocks A and B, each of mass m, are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in Fig. A third identical block C, also of mass m moves on the floor with a speed v along the line joining A and B and collides with A, then
987539_69b591579d70427290d6f69227fb7361.png
  • The KE of the AB system at maximum compression of the spring is zero.
  • The KE of the AB system at maximum compression of the spring is $$(1/4)mv^2$$
  • The maximum compression of the spring is $$v\sqrt{\frac{m}{k}}$$.
  • The maximum compression of the spring is $$v\sqrt{\frac{m}{2k}}$$.
A simple pendulum of length L and mass M is oscillating in a plane about a vertical line between angular limits -$$\phi$$ and +$$\phi$$. For an angular displacement, $$\theta$$(|$$\theta$$| < $$\phi$$) the tension in the string and velocity of the bob are T and v respectively. The following relations hold good under the above condition.
  • T cos $$\theta$$ cos $$\theta$$ = Mg
  • T + Mg cos $$\theta$$ = $$Mv^{2}$$/L
  • The magnitude of tangential acceleration of the bob $$|a_{T}|$$ = g sin $$\theta$$
  • T = Mg (3 cos $$\theta$$ - 2 cos $$\phi$$)
A thin rod of length Land uniform area of cross-section S is pivoted at its lowest point P inside a stationary, homogenous and non-viscous liquid. The rod is free to rotate in a vertical plane about a horizontal axis passing through P. The density $$d_1$$ of the material of the rod is smaller than the density $$d_2$$ of the liquid. The rod is displaced by a small angle $$\theta$$ from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters. 
1010416_5c6103d8387a49c1a4d3e1d2d4ae2e7e.png
  • $$\sqrt{\dfrac{3}{3} \left ( \dfrac{d_2 \, - \, d_1}{d_1} \right ) \dfrac{g}{L}}$$
  • $$\sqrt{\dfrac{3}{4} \left ( \dfrac{d_2 \, - \, d_1}{d_1} \right ) \dfrac{g}{L}}$$
  • $$\sqrt{\dfrac{4}{2} \left ( \dfrac{d_2 \, - \, d_1}{d_1} \right ) \dfrac{g}{L}}$$
  • $$\sqrt{\dfrac{3}{2} \left ( \dfrac{d_2 \, - \, d_1}{d_1} \right ) \dfrac{g}{L}}$$
A particle of mass m is attached to a spring (of spring constant $$k$$) and has a natural angular frequency $$\omega_0$$. An external force $$F(t)$$ proportional to $$\cos \omega t (\omega \neq \omega_0)$$ is applied to the oscillator. The time displacement of the oscillator will be proportional to:
  • $$\dfrac{m}{\omega_0^2-\omega^2}$$
  • $$\dfrac{1}{m(\omega_0^2-\omega^2)}$$
  • $$\dfrac{1}{m(\omega_0^2+\omega^2)}$$
  • $$\dfrac{2}{m(\omega_0^2-\omega^2)}$$
A simple harmonic oscillator of angular frequency 2 rad $$s^{-1}$$ is acted upon by an external force $$F= sin tN.$$ If the oscillator is at rest in its equilibrium position at t=0, its position at later times is proportional to 
  • $$ sin t+ \frac{1}{2}cos 2t$$
  • $$ cos t -\frac{1}{2}sin2t$$
  • $$sint -\frac{1}{2}sin2t$$
  • $$sin t +\frac{1}{2}sin 2t$$
Two $$SHM's$$ with same amplitude and time period, when acting together in perpendicular directions with a phase difference of $$\dfrac{\pi}{2}$$, give rise to
  • Straight motion
  • Elliptical motion
  • Circular motion
  • None of these
Find the angular frequency of small oscillation of block m in the arrangement shown. Rod is massless. [Assume gravity to be absent]
1023909_5e055c8cc3f04b3091ff8da46d1f9171.png
  • $$\sqrt{9k_{1} k_{3} + k_{1}k_{2} + 4k_{2}k_{3}/(4k_{3} + k_{2})m}$$
  • $$\sqrt{4k_{1} k_{3} + k_{1}k_{2} + 4k_{2}k_{3}/(4k_{3} + k_{2})m}$$
  • $$\sqrt{8k_{1} k_{3} + k_{1}k_{2} + 4k_{2}k_{3}/(4k_{3} + k_{2})m}$$
  • $$\sqrt{5k_{1} k_{3} + k_{1}k_{2} + 4k_{2}k_{3}/(4k_{3} + k_{2})m}$$
A particle moves on x-axis according to the equation x = $$x_{0} sin^{2} \omega t$$, the motion is simple harmonic
  • with amplitude $$x_{0}$$
  • with amplitude $$2x_{0}$$
  • with time period $$\displaystyle\frac{2\pi}{\omega}$$
  • with time period $$\displaystyle\frac{\pi}{\omega}$$
STATEMENT-1: Time period of simple harmonic motion at moon is more than that on earth because
STATEMENT-2: There is no atmosphere on moon.
  • Statement-1 is True, Statement-2 is True; Statement -2 is a correct explanation for Statement-1.
  • Statement-1 is True, Statement-2 is True; Statement -2 is NOT a correct explanation for Statement-1.
  • Statement -1 is True, Statement-2 is False.
  • Statement -1 is False, Statement-2 is True.
A highly rigid cubical block A of small mass M and side L is fixed rigidly on to another cubical block B of the same dimensions and of low modulus of rigidity $$\eta$$ such that the lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of a sides faces of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by.
  • $$2\pi\sqrt{M\eta L}$$
  • $$2\pi \sqrt{M\eta /L}$$
  • $$2\pi \sqrt{ML/\eta}$$
  • $$2\pi \sqrt{M/\eta L}$$
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are charged. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along the horizontal axis and momentum is plotted along the `vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $$E_1$$ and $$E_2$$ are the total mechanical energies respectively. Then.
1010225_70ab075a31de456ea62204457aca7ce5.png
  • $$E_1=\sqrt{2}E_2$$
  • $$E_1=2E_2$$
  • $$E_1=4E_2$$
  • $$E_1=16E_2$$
The ratio of the total energies of a oscillating particle and its foot of the perpendicular is
  • 2:1
  • 1:2
  • 1:1
  • $$\sqrt{2}$$:1
Projection of uniform circular motion on  a diameter is:- 
  •  simple harmonic motion
  • angular simple harmonic motion
  • both a and b
  • None of these.
The string of a pendulum is horizontal initially. The mass of the bob attached is m. Now the string is released. 'R' is the length of the pendulum.
(i) Velocity of the string at the lowest point
  • $$\sqrt{2 gR}$$
  • $$\sqrt{3 gR}$$
  • $$\sqrt{4 gR}$$
  • $$\sqrt{5 gR}$$
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 is:
  • $$2\pi \sqrt\frac{V_1^2 - V_2^2}{x_1^2 - x_2^2}$$
  • $$2\pi \sqrt\frac{x_1^2 + x_2^2}{V_1^2 + V_2^2}$$
  • $$2\pi \sqrt\frac{x_1^2 - x_2^2}{V_2^2 - V_1^2}$$
  • $$2\pi \sqrt\frac{V_1^2 + V_2^2}{x_1^2 + x_2^2}$$
The position vector of a particle that is moving in space is given by $$\vec {r} =(1 +2\cos 2\omega t)\hat {i} + (3\sin^{2} \omega t)\hat {j} + (3)\hat {k}$$ in the ground frame. All units are in $$SI$$. Choose the correct statement(s).
  • The particle executes SHM in the ground frame about the mean position $$\left (1, \dfrac {3}{2}, 3\right )$$
  • The particle executes SHM in a frame moving along the z-axis with a velocity of $$3\ m/s$$
  • The amplitude of the SHM of the particle is $$\dfrac {5}{2} m$$
  • The direction of the SHM of the particle is given by the vector $$\left (\dfrac {4}{5} \hat {i} - \dfrac {3}{5} \hat {j}\right )$$
The string of a pendulum is horizontal initially. The mass of the bob attached is m. Now the string is released. 'R' is the length of the pendulum.
(i) The tension in the string at an angle $$30 ^\circ$$ with the vertical is ______
  • $$2 \sqrt{5} mg$$
  • $$2 \sqrt{3} mg$$
  • $$\dfrac{3 \sqrt{3}}{2} mg$$
  • $$4 \sqrt 3 mg$$

The maximum value attained by the tension in the string of a
swinging pendulum is four times the minimum value it attains. There is no stack
in the string. The angular amplitude of the pendulum is
  • $${90^0}$$
  • $${60^0}$$
  • $${45^0}$$
  • $${30^0}$$
A simple pendulum with a brass bob has a time period T. The bob is now immersed in a non-viscous liquid and made to oscillate. The density of the liquid is (1/8)th that of the brass. The time period of the pendulum will  be 
  • $$\sqrt{\dfrac{8}{7}}$$ g
  • $$\dfrac{7}{8}$$ g
  • $$\dfrac{8^2}{7^2}$$ g
  • g
Two pendulums of lengths 121 cm and 100 cm start vibrating at the same instant. They are in the mean position and in the same phase. After how many vibrations of the shorter pendulum, the two will be in the same phase in the mean position? 
  • 10 vibrations
  • 11 vibrations
  • 21 vibrations
  • 20 vibrations
The equation for the displacement of a particle executing SHM is $$y={5 \sin 2 \pi t}$$ cm . Find (i) the velocity at 3cm from the mean position,(ii) acceleration after 0.5s after leaving the mean position.
  • $$8 \pi cms^{-1},zero$$
  • $$6 \pi cms^{-1},zero$$
  • $$4 \pi cms^{-1},zero$$
  • $$2 \pi cms^{-1},zero$$
A particle executes SHM with amplitude $$0.5\ cm$$ and frequency $${100s}^{-1}$$. The maximum speed of the particle is (in m/s)
  • $$\pi $$
  • $$0.5$$
  • $$5\pi \times{10}^{-5}$$
  • $$100\pi$$
A heavy bass-sphere is hung from a spiral spring and it executes vertical vibrations with period $$T$$. The ball is now immersed in nonviscous liquid with a density one-tenth that of brass. When set into vertical vibrations with the sphere remaining inside the liquid all the time, the period will be-
  • $$10\left( { 3 }^{ 1/2 } \right) cm\left[ \dfrac { 9 }{ 10 } \right] T$$
  • $$T\sqrt { \left( \dfrac { 10 }{ 9 } \right) }$$
  • Unchanged
  • $$T\sqrt { \left( \dfrac { 9 }{ 10 } \right) }$$
A body is projected with an initial velocity $$20 \ m/s$$ at $$60^\circ$$ to the horizontal. The displacement after $$2 s$$ is:
  • $$20 [i + (\sqrt{3} - 1) j]$$
  • $$20 [i - (\sqrt{3} - 1) j]$$
  • $$10 [i + (\sqrt{3} + 1) j]$$
  • $$10 [i + (\sqrt{3} - 1) j]$$
Find if the following functions of time are
(a) Periodic but not SHM
  • $$\sin \omega t+ \sin \omega t+ \cos 2 \omega t$$
  • $$e^{-\omega t}$$
  • $$\log (\omega t)$$
  • $$\sin^{3} \omega t$$
What of the following quantities are always positive in a SHM
  • $$\bar{F}.\bar{a}$$
  • $$\bar{a}.\bar{r}$$
  • $$\bar{v}.\bar{r}$$
  • $$\bar{F}.\bar{r}$$
The system shown in lying on a smooth horizontal surface. Mass $$m$$ is constrained to move along the line $$3$$. Springs $$1$$ and $$2$$ at right angles with respect to one another are symmetrically arranged with respect to $$3$$ which is an elastic cord. The force constants of $$1,2$$ and $$3$$ are $$k$$ each and they are all just taut in the condition shown. For small oscillations of $$m$$ along $$3$$, time period is
1120032_8e607186719549369bf9c49386f79a1a.png
  • $$\pi \sqrt { \dfrac { m }{ 2k } } \left( 1+\sqrt { 2 } \right) $$
  • $$2\pi \sqrt { \dfrac { m }{ 2k } } \left( 1+\sqrt { 2 } \right) $$
  • $$\dfrac {3\pi}{2} \sqrt { \dfrac { m }{ 2k } } \left( 1+\sqrt { 2 } \right) $$
  • $$\pi \sqrt { \dfrac { 2m }{ k } } \left( 1+\sqrt { 2 } \right) $$
For a particle performing linear S.H.M. its average speed over one oscillation is where $$A$$ is amplitude of S.H.M.
  • Zero
  • $$\dfrac {A\omega}{\pi}$$
  • $$\dfrac {A\omega}{2\pi}$$
  • $$\dfrac {2A\omega}{\pi}$$
Two masses $${m_1}$$ and $${m_2}$$ are joined by a spring as shown. The system is dropped to the ground from a certain height. The spring will be :-
1206449_9d46c4c8157045c590deb7593ce23adb.png
  • Stretched when $${m_2} > {m_1}$$
  • compressed when $${m_2} < {m_1}$$
  • neither compressed nor stretched only when $${m_1} = {m_2}$$
  • neither compressed nor stretched regardless of the values of $${m_1}$$ and $${m_2}$$
A horizontal force $$F$$ is applied to the lower end of a uniform thin rod of mass $$4kg$$ and length $$L = 50\ cm$$ as shown in the figure. The rod undergoes only translational motion along the smooth horizontal surface. If $$F = 60\ N$$ determine the angle $$\theta$$ for translation motion of the rod. $$(g =10\ m/s^{2})$$.
1208123_2d7e71bdcdc148d78569548e234e44d5.png
  • $$\tan^{-1} \left (\dfrac {2}{3}\right )$$
  • $$\tan^{-1} \left (\dfrac {1}{2}\right )$$
  • $$\sin^{-1} \left (\dfrac {1}{5}\right )$$
  • $$\sin^{-1} \left (\dfrac {1}{4}\right )$$
Two particle execute SHM with same amplitude 'A' and same angular frequency along same line and about same mean position. During oscillation they cross each other in opposite direction at a distance $$\frac{{\sqrt 3 }}{2}A$$ from mean position. The phase difference of two SHM is - 
  • $$\pi$$ / 2
  • $$\pi$$ / 3
  • 2$$\pi$$ / 3
  • 5$$\pi$$ / 6
Clock A based on spring oscillations and a clock B based on oscillations of simple pendulum are synchronized on earth. Both are taken to mars whose mass is 0.1 times the mass of earth and radius is half that of earth. Which of the following statement is correct
  • Both will show same time
  • Time measured in clock A will be greater than that in clock B
  • Time measured in clock B will be greater than that in clock A
  • Clock A will stop and clock B will show time as it shows on earth
Two particles are executing identical simple harmonic motions described by the equations, $$x_1=a\cos \left(\omega t+\left(\dfrac{\pi}{6}\right)\right)$$ and $$x_2=a\cos \left(\omega t+\dfrac{\pi}{3}\right)$$. The minimum interval of time between the particles crossing the respective mean position is?
  • $$\dfrac{\pi}{2\omega}$$
  • $$\dfrac{\pi}{3\omega}$$
  • $$\dfrac{\pi}{4\omega}$$
  • $$\dfrac{\pi}{6\omega}$$
The potential energy of a particle executing SHM along the x-axis is given by $$U=U_0-U_0\cos ax$$. What is the period of oscillation?
  • $$2\pi\sqrt{\dfrac{ma}{U_o}}$$
  • $$2\pi\sqrt{\dfrac{U_o}{ma}}$$
  • $$\dfrac{2\pi}{a}\sqrt{\dfrac{m}{U_o}}$$
  • $$2\pi\sqrt{\dfrac{m}{aU_o}}$$
A pendulum comprising a light string of length $$L$$ and a small sphere, swing in the vertical plane. The string hits a peg located a distance $$d$$ below the point of suspension as shown in figure. If the pendulum is released from the horizontal position ($$\theta = 90^o$$) and is to swing in a complete circle centered on the peg, then the minimum value of $$d$$ is
1193977_580802ec93c44aac8fac6bb503bcfd1a.png
  • $$L/5$$
  • $$2L/5$$
  • $$4L/5$$
  • $$3L/5$$
uniform rod of mass m and length I is suspended about its end. Time period is

1189601_41cd20718d574ab294a424a9b963dfb4.jpg
  • $$2 \pi \sqrt { \cfrac { l } { g } }$$
  • $$2 \pi \sqrt { \cfrac { 2l } { g } }$$
  • $$2 \pi \sqrt { \cfrac { 2 l } { 3 g } }$$
  • $$2 \pi \sqrt { \cfrac { l } { 3 g } }$$
The kind of motion that a pendulum shows 
  • circular motion
  • rectilinear motion
  • oscillatory motion
  • randam motion
A block of mass $$m$$ is suspended from one end of a light spring as shown. The origin $$O$$ is considered at distance equal to natural length of spring from ceiling and vertical downward direction as positive $$Y-axis.$$ When the system is in equilibrium a bullet of mass $$m / 3$$ moving in vertical upward direction with velocity $$v _ { 0 }$$ strikes the block and embeds into it. As a result, the block (with bullet embedded into it) moves up and starts oscillating.
Mark out the correct statements:
1191267_5c7c8198fc8246d1aedd7ec4f47652b2.jpg
  • The block-bullet system performs SHM about
    $$y = \frac { m g } { k } .$$
  • The block-bullet system performs oscillatory motion
    but not SHM about $$y = \frac { m g } { k }.$$
  • The block-bullet system performs SHM about
    $$y = \frac { 4 m g } { 3 k }.$$
  • The block-bullet system performs oscillatory motion
    but not SHM about $$y = \frac { 4 m g } { 3 k }.$$
A particle is performing S.H.M. and at $$t = \dfrac{3T}{4}$$, is at position = $$\dfrac{A}{\sqrt 2}$$ and moving towards the origin. Equilibrium position of the particle is at $$x = 0$$. After $$t = \dfrac{3T}{2}$$ what will be the graph of the particle :-
The differential equation representing the SHM of a particule is $$\cfrac {{9d^2y}} {dt^2} +4Y=0.$$The time period of the particle is fiven by
  • $${\cfrac \pi 3}sec$$
  • $$ \pi sec$$
  • $${\cfrac {2\pi} 3}sec$$
  • $${3\pi }sec$$
Two point masses of 3.0 kg and 6.0 kg are attached to opposite ends of horizontal sprig whose spring constant is 300 N$$m^{-1}$$ as shown in the figure. The natural vibration frequency of the system is approximately.
1217869_357fdc71631e40109046c7edb44c8eef.PNG
  • 4 Hz
  • 3 Hz
  • 2 Hz
  • 1 Hz

A steamer moves with velocity 3 km/h in and against the direction of river water whose velocity is 2 km/h. Calculate the total time for the total journey if the boat travels 2 km in the direction of a stream and then back to its place:
  • 2 hrs
  • 2.5 hrs
  • 2.4 hrs
  • 3 hrs.
A point performs simple harmonic oscillation of period $$T$$ and the equation of motion is given by $$x=a\sin{(\omega+\pi/6)}$$. After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity ?
  • $$T/3$$
  • $$T/12$$
  • $$T/8$$
  • $$T/6$$
A block of mass $$M$$ is attached with the springs as shown .If the block is slightly displaced the time period of $$SHM$$ for the block shown in the figure will be
1228998_c0ead2b0b9024d10baabbc974314c95d.png
  • $$ 2 \pi\sqrt {\dfrac{m}{9k}}$$
  • $$ 2 \pi\sqrt {\dfrac{m}{k}}$$
  • $$ \dfrac{4 \pi}{3} \sqrt {\dfrac{m}{9k}}$$
  • $$ \pi\sqrt {\dfrac{m}{2k}}$$
At some instant, velocity and acceleration of a particle of mass '$$m$$' in SHM are '$$a$$' and '$$b$$' respectively. If its angular frequency is $$\omega,$$ then its maximum K.E. is
  • $$\frac { 1 } { 2 } m \left[ a ^ { 2 } + \frac { b ^ { 2 } } { \omega } \right]$$
  • $$\frac { 1 } { 2 } m \left[ a ^ { 2 } - \frac { b ^ { 2 } } { \omega ^ { 2 } } \right]$$
  • $$\frac { 1 } { 2 } m \left[ a ^ { 2 } + \frac { b ^ { 2 } } { \omega ^ { 2 } } \right]$$
  • $$\frac { 1 } { 2 } m \left[ a ^ { 2 } - \frac { b ^ { 2 } } { \omega } \right]$$
An object of mass m is tied to a string of length l and a veriable horizontal force is appiled on it, which starts at zero and gradully increases (it is pulled extremely slowly so that equilibrium exists al all times) untill the string makes an angle $$\theta$$ with the vertical. Work done by the force F is : 
1223918_048fab91b8a348e1a3c80148991bc4f2.png
  • $$mgl (1 - \sin { \theta } $$)
  • $$mgl$$
  • $$mgl (1 - \cos { \theta }$$)
  • $$mgl (1 - \tan {\theta }$$)
The equation of motion of a particle executing S.H.M. where letters have usual meaning is:
  • $$\dfrac{d^2x}{dt^2}=\dfrac{k}{m}x$$
  • $$\dfrac{d^2x}{dt^2}=+\omega^2x$$
  • $$\dfrac{d^2x}{dt^2}=-\omega^2x^2$$
  • $$\dfrac{d^2x}{dt^2}=-kmx$$
A source of sound is moving along a circular orbit of radius $$3\ m$$ with an angular velocity of $$10$$ rads/s. A sound detector located for away from the source is executing SHM along the line BD with amplitude $$BC=CD=6\ m$$. The frequency of oscillation of the detector is $$\cfrac{5}{\pi}$$ per sec. The source is at the point $$A$$ when the detector is at the point $$B$$. If the source emits a continuous sound wave frequency $$340$$ Hz. Find the ratio of maximum to minimum frequencies recorded by the detector. (velocity of sound $$= 330\ m/s$$)
1276974_3ec0a656744e460f8e8af19733c5524d.png
  • $$\cfrac{442}{255}$$
  • $$\cfrac{255}{442}$$
  • $$\cfrac{221}{89}$$
  • $$\cfrac{442}{300}$$
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