
Two particles executes S.H.M. of same amplitude and frequency along the same straight line. They pass one another when going in opposite directions, and each time their displacement is half of their amplitude. The phase difference between them is [MP PMT 1999]
A)
30° done
clear
B)
60° done
clear
C)
90° done
clear
D)
120° done
clear
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The displacement of a particle varies with time as \[x=12\sin \omega t16{{\sin }^{3}}\omega t\](in cm). If its motion is S.H.M., then its maximum acceleration is
A)
\[12\,{{\omega }^{2}}\] done
clear
B)
\[36\,{{\omega }^{2}}\] done
clear
C)
\[144\,{{\omega }^{2}}\] done
clear
D)
\[\sqrt{192}\,{{\omega }^{2}}\] done
clear
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A linear harmonic oscillator of force constant \[2\times {{10}^{6}}N/m\]times and amplitude 0.01 m has a total mechanical energy of 160 joules. Its [IIT JEE 1989; CPMT 1995; CBSE PMT 1996; KECT (Med.) 1999; AMU (Engg.) 2000; UPSEAT 2001]
A)
Maximum potential energy is 100 J done
clear
B)
Maximum K.E. is 100 J done
clear
C)
Maximum P.E. is 160 J done
clear
D)
Minimum P.E. is zero done
clear
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A particle of mass m is executing oscillations about the origin on the xaxis. Its potential energy is \[U(x)=k{{[x]}^{3}}\], where k is a positive constant. If the amplitude of oscillation is a, then its time period T is [IITJEE 1998]
A)
Proportional to \[\frac{1}{\sqrt{a}}\] done
clear
B)
Independent of a done
clear
C)
Proportional to \[\sqrt{a}\] done
clear
D)
Proportional to \[{{a}^{3/2}}\] done
clear
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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 figure. 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 [IITJEE 1993]
A)
The kinetic energy of the AB system at maximum compression of the spring is zero done
clear
B)
The kinetic energy of the AB system at maximum compression of the spring is \[m{{v}^{2}}/4\] done
clear
C)
The maximum compression of the spring is \[v\sqrt{m/K}\] done
clear
D)
The maximum compression of the spring is \[v\sqrt{m/2K}\] done
clear
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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 [IITJEE 1981]
A)
\[T=2\pi \sqrt{\left( \frac{Mh}{PA} \right)}\] done
clear
B)
\[T=2\pi \sqrt{\left( \frac{MA}{Ph} \right)}\] done
clear
C)
\[T=2\pi \sqrt{\left( \frac{M}{PAh} \right)}\] done
clear
D)
\[T=2\pi \sqrt{MPhA}\] done
clear
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A sphere of radius r is kept on a concave mirror of radius of curvature R. The arrangement is kept on a horizontal table (the surface of concave mirror is frictionless and sliding not rolling). If the sphere is displaced from its equilibrium position and left, then it executes S.H.M. The period of oscillation will be
A)
\[2\pi \sqrt{\left( \frac{\left( Rr \right)1.4}{g} \right)}\] done
clear
B)
\[2\pi \sqrt{\left( \frac{Rr}{g} \right)}\] done
clear
C)
\[2\pi \sqrt{\left( \frac{rR}{a} \right)}\] done
clear
D)
\[2\pi \sqrt{\left( \frac{R}{gr} \right)}\] done
clear
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The amplitude of vibration of a particle is given by \[{{a}_{m}}=({{a}_{0}})/(a{{\omega }^{2}}b\omega +c);\] where \[{{a}_{0}},a,b\] and c are positive. The condition for a single resonant frequency is [CPMT 1982]
A)
\[{{b}^{2}}=4ac\] done
clear
B)
\[{{b}^{2}}>4ac\] done
clear
C)
\[{{b}^{2}}=5ac\] done
clear
D)
\[{{b}^{2}}=7ac\] done
clear
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A U tube of uniform bore of crosssectional area A has been set up vertically with open ends facing up. Now m gm of a liquid of density d is poured into it. The column of liquid in this tube will oscillate with a period T such that
A)
\[T=2\pi \sqrt{\frac{M}{g}}\] done
clear
B)
\[T=2\pi \sqrt{\frac{MA}{gd}}\] done
clear
C)
\[T=2\pi \sqrt{\frac{M}{gdA}}\] done
clear
D)
\[T=2\pi \sqrt{\frac{M}{2Adg}}\] done
clear
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A particle is performing simple harmonic motion along xaxis with amplitude 4 cm and time period 1.2 sec. The minimum time taken by the particle to move from x =2 cm to x = + 4 cm and back again is given by [AIIMS 1995]
A)
0.6 sec done
clear
B)
0.4 sec done
clear
C)
0.3 sec done
clear
D)
0.2 sec done
clear
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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 [SCRA 1994; AIIMS 1995]
A)
0.5 Hz done
clear
B)
1.5 Hz done
clear
C)
5 Hz done
clear
D)
10 Hz done
clear
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Due to some force F1 a body oscillates with period 4/5 sec and due to other force F2 oscillates with period 3/5 sec. If both forces act simultaneously, the new period will be [RPET 1997]
A)
0.72 sec done
clear
B)
0.64 sec done
clear
C)
0.48 sec done
clear
D)
0.36 sec done
clear
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A horizontal platform with an object placed on it is executing S.H.M. in the vertical direction. The amplitude of oscillation is \[3.92\times {{10}^{3}}m\]. What must be the least period of these oscillations, so that the object is not detached from the platform [AIIMS 1999]
A)
0.1256 sec done
clear
B)
0.1356 sec done
clear
C)
0.1456 sec done
clear
D)
0.1556 sec done
clear
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A particle executes simple harmonic motion (amplitude = A) between \[x=A\] and \[x=+A\]. The time taken for it to go from 0 to A/2 is \[{{T}_{1}}\] and to go from A/2 to A is \[{{T}_{2}}\]. Then [IITJEE (Screening) 2001]
A)
\[{{T}_{1}}<{{T}_{2}}\] done
clear
B)
\[{{T}_{1}}>{{T}_{2}}\] done
clear
C)
\[{{T}_{1}}={{T}_{2}}\] done
clear
D)
\[{{T}_{1}}=2{{T}_{2}}\] done
clear
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A simple pendulum of length L and mass (bob) M is oscillating in a plane about a vertical line between angular limits \[\varphi \] and\[+\varphi \]. For an angular displacement\[\theta (\theta <\varphi )\], the tension in the string and the velocity of the bob are T and v respectively. The following relations hold good under the above conditions [IIT 1986; UPSEAT 1998]
A)
\[T\cos \theta =Mg\] done
clear
B)
\[TMg\cos \theta =\frac{M{{v}^{2}}}{L}\] done
clear
C)
The magnitude of the tangential acceleration of the bob \[{{a}_{T}}\,=g\sin \theta \] done
clear
D)
\[T=Mg\cos \theta \] done
clear
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Two simple pendulums of length 5 m and 20 m respectively are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has completed .... oscillations. [CBSE PMT 1998; JIPMER 2001, 02]
A)
5 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
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The bob of a simple pendulum is displaced from its equilibrium position O to a position Q which is at height h above O and the bob is then released. Assuming the mass of the bob to be m and time period of oscillations to be 2.0 sec, the tension in the string when the bob passes through O is [AMU 1995]
A)
\[m\,(g+\pi \sqrt{2g\,h})\] done
clear
B)
\[m\,(g+\sqrt{{{\pi }^{2}}g\,h})\] done
clear
C)
\[m\,\left( g+\sqrt{\frac{{{\pi }^{2}}}{2}g\,h} \right)\] done
clear
D)
\[m\,\left( g+\sqrt{\frac{{{\pi }^{2}}}{3}g\,h} \right)\] done
clear
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The metallic bob of a simple pendulum has the relative densityr. The time period of this pendulum is T. If the metallic bob is immersed in water, then the new time period is given by [SCRA 1998]
A)
\[T\frac{\rho 1}{\rho }\] done
clear
B)
\[T\frac{\rho }{\rho 1}\] done
clear
C)
\[T\sqrt{\frac{\rho 1}{\rho }}\] done
clear
D)
\[T\sqrt{\frac{\rho }{\rho 1}}\] done
clear
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A clock which keeps correct time at \[{{20}^{o}}C\], is subjected to \[{{40}^{o}}C\]. If coefficient of linear expansion of the pendulum is \[12\times {{10}^{6}}/{}^\circ C\]. How much will it gain or loose in time [BHU 1998]
A)
10.3 seconds / day done
clear
B)
20.6 seconds / day done
clear
C)
5 seconds / day done
clear
D)
20 minutes / day done
clear
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The period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination a, is given by [IITJEE (Screening) 2000]
A)
\[2\pi \sqrt{\frac{L}{g\cos \alpha }}\] done
clear
B)
\[2\pi \sqrt{\frac{L}{g\sin \alpha }}\] done
clear
C)
\[2\pi \sqrt{\frac{L}{g}}\] done
clear
D)
\[2\pi \sqrt{\frac{L}{g\tan \alpha }}\] done
clear
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The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is \[{{t}_{0}}\]in air. Neglecting frictional force of water and given that the density of the bob is (4/3) ×1000 kg/m3. What relationship between \[t\]and \[{{t}_{0}}\]is true [AIEEE 2004]
A)
\[t={{t}_{0}}\] done
clear
B)
\[t={{t}_{0}}/2\] done
clear
C)
\[t=2{{t}_{0}}\] done
clear
D)
\[t=4{{t}_{0}}\] done
clear
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A spring of force constant k is cut into two pieces such that one piece is double the length of the other. Then the long piece will have a force constant of [IITJEE (Screening) 1999]
A)
\[(2/3)k\] done
clear
B)
\[(3/2)k\] done
clear
C)
\[3k\] done
clear
D)
\[6k\] done
clear
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One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of crosssection and Young's modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to [IIT 1993]
A)
\[2\pi \left( \frac{m}{K} \right)\] done
clear
B)
\[2\pi {{\left\{ \frac{(YA+KL)m}{YAK} \right\}}^{1/2}}\] done
clear
C)
\[2\pi \frac{mYA}{KL}\] done
clear
D)
\[2\pi \frac{mL}{YA}\] done
clear
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On a smooth inclined plane, a body of mass M is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has force constant K, the period of oscillation of the body (assuming the springs as massless) is [NSEP 1994]
A)
\[2\pi {{\left( \frac{m}{2K} \right)}^{1/2}}\] done
clear
B)
\[2\pi {{\left( \frac{2M}{K} \right)}^{1/2}}\] done
clear
C)
\[2\pi \frac{Mg\sin \theta }{2K}\] done
clear
D)
\[2\pi {{\left( \frac{2Mg}{K} \right)}^{1/2}}\] done
clear
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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 \[\cos \omega \,t((\omega \ne {{\omega }_{0}})\]is applied to the oscillator. The time displacement of the oscillator will be proportional to [AIEEE 2004]
A)
\[\frac{m}{\omega _{0}^{2}{{\omega }^{2}}}\] done
clear
B)
\[\frac{1}{m(\omega _{0}^{2}{{\omega }^{2}})}\] done
clear
C)
\[\frac{1}{m(\omega _{1}^{2}+{{\omega }^{2}})}\] done
clear
D)
\[\frac{m}{\omega _{1}^{2}+{{\omega }^{2}}}\] done
clear
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A 15 g ball is shot from a spring gun whose spring has a force constant of 600 N/m. The spring is compressed by 5 cm. The greatest possible horizontal range of the ball for this compression is (g = 10 m/s2) [DPMT 2004]
A)
6.0 m done
clear
B)
10.0 m done
clear
C)
12.0 m done
clear
D)
8.0 m done
clear
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An ideal spring with springconstant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is [IITJEE (Screening) 2002]
A)
4 Mg/K done
clear
B)
2 Mg/K done
clear
C)
Mg/K done
clear
D)
Mg/2K done
clear
View Solution play_arrow

The displacement y of a particle executing periodic motion is given by\[y=4{{\cos }^{2}}(t/2)\sin (1000t)\]. This expression may be considered to be a result of the superposition of........... independent harmonic motions [IIT 1992]
A)
Two done
clear
B)
Three done
clear
C)
Four done
clear
D)
Five done
clear
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Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by \[{{45}^{o}}\], then [IIT JEE 1999]
A)
The resultant amplitude is \[(1+\sqrt{2)}a\] done
clear
B)
The phase of the resultant motion relative to the first is 90° done
clear
C)
The energy associated with the resulting motion is \[(3+2\sqrt{2)}\] times the energy associated with any single motion done
clear
D)
The resulting motion is not simple harmonic done
clear
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The function \[{{\sin }^{2}}(\omega t)\]represents [AIEEE 2005]
A)
A simple harmonic motion with a period \[2\pi /\omega \] done
clear
B)
A simple harmonic motion with a period \[\pi /\omega \] done
clear
C)
A periodic but not simple harmonic motion with a period \[2\pi /\omega \] done
clear
D)
A periodic but not simple harmonic, motion with a period \[\pi /\omega \] done
clear
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A simple pendulum has time period T1. The point of suspension is now moved upward according to equation \[y=k{{t}^{2}}\] where\[k=1\,m/se{{c}^{2}}\]. If new time period is T2 then ratio \[\frac{T_{1}^{2}}{T_{2}^{2}}\] will be [IITJEE (Screening) 2005]
A)
2/3 done
clear
B)
5/6 done
clear
C)
6/5 done
clear
D)
3/2 done
clear
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A simple pendulum is hanging from a peg inserted in a vertical wall. Its bob is stretched in horizontal position from the wall and is left free to move. The bob hits on the wall the coefficient of restitution is \[\frac{2}{\sqrt{5}}\]. After how many collisions the amplitude of vibration will become less than 60° [UPSEAT 1999]
A)
6 done
clear
B)
3 done
clear
C)
5 done
clear
D)
4 done
clear
View Solution play_arrow

A brass cube of side a and density s is floating in mercury of densityr. If the cube is displaced a bit vertically, it executes S.H.M. Its time period will be
A)
\[2\pi \sqrt{\frac{\sigma \,a}{\rho \,g}}\] done
clear
B)
\[2\pi \sqrt{\frac{\rho \,a}{\sigma \,g}}\] done
clear
C)
\[2\pi \sqrt{\frac{\rho \,g}{\sigma \,a}}\] done
clear
D)
\[2\pi \sqrt{\frac{\sigma \,g}{\rho \,a}}\] done
clear
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Two identical balls A and B each of mass 0.1 kg are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in the figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius 0.06 m. Each spring has a natural length of 0.06p m and force constant 0.1N/m. Initially both the balls are displaced by an angle \[\theta =\pi /6\] radian with respect to the diameter \[PQ\] of the circle and released from rest. The frequency of oscillation of the ball B is
A)
\[\pi \,Hz\] done
clear
B)
\[\frac{1}{\pi }Hz\] done
clear
C)
\[2\pi \,Hz\] done
clear
D)
\[\frac{1}{2\pi }Hz\] done
clear
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A disc of radius R and mass M is pivoted at the rim and is set for small oscillations. If simple pendulum has to have the same period as that of the disc, the length of the simple pendulum should be
A)
\[\frac{5}{4}R\] done
clear
B)
\[\frac{2}{3}R\] done
clear
C)
\[\frac{3}{4}R\] done
clear
D)
\[\frac{3}{2}R\] done
clear
View Solution play_arrow

One end of a spring of force constant k is fixed to a vertical wall and the other to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance \[{{x}_{0}}\] from the black. The spring is then compressed by \[2{{x}_{0}}\] and released. The time taken to strike the wall is
A)
\[\frac{1}{6}\pi \sqrt{\frac{k}{m}}\] done
clear
B)
\[\sqrt{\frac{k}{m}}\] done
clear
C)
\[\frac{2\pi }{3}\sqrt{\frac{m}{k}}\] done
clear
D)
\[\frac{\pi }{4}\sqrt{\frac{k}{m}}\] done
clear
View Solution play_arrow

Three masses 700g, 500g, and 400g are suspended at the end of a spring a shown and are in equilibrium. When the 700g mass is removed, the system oscillates with a period of 3 seconds, when the 500 gm mass is also removed, it will oscillate with a period of
A)
1 s done
clear
B)
2 s done
clear
C)
3 s done
clear
D)
\[\sqrt{\frac{12}{5}}\,s\] done
clear
View Solution play_arrow

A particle of mass m is attached to three identical springs A, B and C each of force constant k a shown in figure. If the particle of mass m is pushed slightly against the spring A and released then the time period of oscillations is
A)
\[2\pi \sqrt{\frac{2m}{k}}\] done
clear
B)
\[2\pi \sqrt{\frac{m}{2k}}\] done
clear
C)
\[2\pi \sqrt{\frac{m}{k}}\] done
clear
D)
\[2\pi \sqrt{\frac{m}{3k}}\] done
clear
View Solution play_arrow

A hollow sphere is filled with water through a small hole in it. It is then hung by a long thread and made to oscillate. As the water slowly flows out of the hole at the bottom, the period of oscillation will [MP PMT 1994; KCET 1994; RPET 1996; AFMC 2000; CBSE PMT 2000; CPMT 2001; AIEEE 2005]
A)
Continuously decrease done
clear
B)
Continuously increase done
clear
C)
First decrease and then increase to original value done
clear
D)
First increase and then decrease to original value done
clear
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Two simple pendulums whose lengths are 100 cm and 121 cm are suspended side by side. Their bobs are pulled together and then released. After how many minimum oscillations of the longer pendulum, will the two be in phase again [DPMT 2005]
A)
11 done
clear
B)
10 done
clear
C)
21 done
clear
D)
20 done
clear
View Solution play_arrow

The amplitude of a damped oscillator becomes half in one minute. The amplitude after 3 minute will be \[\frac{1}{X}\]times the original, where X is [CPMT 1989; DPMT 2002]
A)
\[2\times 3\] done
clear
B)
\[{{2}^{3}}\] done
clear
C)
\[{{3}^{2}}\] done
clear
D)
\[3\times {{2}^{2}}\] done
clear
View Solution play_arrow

Which of the following function represents a simple harmonic oscillation [AIIMS 2005]
A)
\[\sin \omega t\cos \omega t\] done
clear
B)
\[{{\sin }^{2}}\omega t\] done
clear
C)
\[\sin \omega x+\sin 2\omega t\] done
clear
D)
\[\sin \omega x\sin 2\omega t\] done
clear
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A uniform rod of length 2.0 m is suspended through an end and is set into oscillation with small amplitude under gravity. The time period of oscillation is approximately [AMU (Med.) 2000]
A)
1.60 sec done
clear
B)
1.80 sec done
clear
C)
2.0 sec done
clear
D)
2.40 sec done
clear
View Solution play_arrow