### For JEE Main other Engineering Entrance Exam Preparation, **JEE Main Physics Simple Harmonic Motion** Previous Year Questions with Solutions is given below.

**Multiple Choice with ONE correct answer**

**1.Two bodies M and N of equal masses are suspended from two separate massless springs of spring constants k1 and k _{2} respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of M to that of N is [1988-1mark]**

**Ans.**

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**3.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 [2000-2 marks]**

**Ans.**

**4.A particle executes simple harmonic motion between x = -A and x = +A. The time taken for it to go from 0 to A/2 is T _{t} and to go from A/2 to A is T_{2}. Then [2001-2 marks]**

**a) T1 < T**

_{2}b) T1 > T_{2}c)T1 = T_{2}d) T1 = 2T_{2}**Ans.**

Option (a) represents the correct answer. Infact, velocity of particle goes on decreasing from maximum value to zero as, the particle travels from mean position to extreme position.

**Ans.**

**6.A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum, its acceleration vector is a correctly shown in [2002]**

**Ans.**

**7.(i) For a particle executing SHM the displacement x is given by x=A cos ut, Identify the graph which represents the variation of potential energy (PE) as a function of time t and displacement x**

**a)5/6 b)6/5 c)1 d)4/5**

**Ans.**

**8.A block (B) is attached to two unstretched springs Sj and S _{2} with spring constants k and 4k, respectively (see figure 1). The other ends are attached to identical supports Mj and M_{2} not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio y/x is**

**a)4 b)2 c)1/2 d)1/4**

**Ans.**(c) When the block B is displaced to the right, the spring S

_{2}will have no tension, it will be in the natural length. And when the block B is displaced to the left. The spring S, will have no tension, it will be in its natural length.

**9.The x-t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at t = 4/3s is**

**Ans.**

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**11.The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is [2009]**

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**13.A wooden block performs SHM on a frictionless surface with frequency, v _{0}. The block carries a charge +Q on its surface. If now a uniform electric field E is switched – on as shown, then the SHM of the block will be E**

**a)of the same frequency and with shifted mean positio**

**b)of the same frequency and with the same mean position**

**c)of changed frequency and with shifted mean position**

**d)of changed frequency and with the same mean position**

**Ans.**

**Ans.**

**Multiple Choice with ONE or More Than ONE correct answers**

**15.A particle executes simple harmonic motion with a frequency f. The frequency with which its kinetic energy oscillates is [1987-2 marks]**

**a) f/2 b) f c) 2f d) 4f**

**Ans.**(c) During one complete oscillation, the kinetic energy of particle executing simple harmonic motion will become maximum twice. It becomes 2f.

Frequency of oscillation of K.E. = 2f.

**16.A linear harmonic oscillator of force constant 2 x 10 ^{6} N/m and amplitude 0.01m has a total mechanical energy of 160J. Its [1989-2 marks]**

**a) maximum potential energy is 100 J b) maximum kinetic energy is 100 J**

**c) maximum potential energy is 160 J d) maximum potential energy is zero**

**Ans.**

**17.A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half-submerged in a liquid of density p at equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with small amplitude. If the force constant of the spring is k, the frequency of oscillation of the cylinder is [1990-2 marks]**

**Ans.**

**18.A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of the same dimensions and of low modulus of rigidity n 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 the side faces of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by [1992-2 marks]**

**Ans.**(d)The cubical block A is placed above cubical block B. They have same dimensions. The lower face of B is held rigidly or a horizontal surface. A force F is applied to the upper cube A at right angles to one of the side faces. The block A executes SHM when the force is withdrawn. The lower block gets distorted.

**19.One end of a long metallic wire of length L’Is tied to the ceiling. The other end is tied to a massless spring 6f spring constant k. A mass m hangs freely from the free end of the spring. The area of cross section and the 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**

**Ans.**

**20.A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is V(x) = k|x|**^{3}** where k is a positive constant. If the amplitude of oscillations is a, then its time period T is [1998-2 marks]**

**Ans.**

**21.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°, then 1999-3 marks]**

**Ans.**

**Ans.**

**Ans.**

**24.A metal rod of length ‘L’ and mass ‘m’ is pivoted at one end. A thin disk of mass ‘M’ and radius ‘R’(< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached; (case A) the disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod – disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true ? [2011]**

**a)Restoring torque in case A = Restoring torque in case B**

**b)Restoring torque in case A < Restoring torque in case B**

**c)Angular frequency for case A > Angular frequency for case B**

**d)Angular frequency for case A < Angular frequency for case B**

**Ans.**

**Comprehension based question**

**25.If the total energy of the particle is E, it will perform periodic motion only if**

**a)E < 0 b) E > 0**

**c) V _{0} > E > 0 d) E > V_{0}**

**Ans.C**

**26.For periodic motion of small amplitude A, the time period T of this particle is proportional to**

**Ans.(b)**

**27.acceleration of this particle for |x|>x _{0} is**

**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 changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along 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**

**Ans.**

**28.The phase space diagram for a ball thrown vertically up from ground is**

**Ans.**(d) Vertically thrown particle comes back to original position till the particle the highest position of its path, its momentum is +ve. After, it become zero and then negative

**29.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 E1 and E**_{2 }**are the total mechanical energies respectively. Then [2011]**

**Ans.**

**30.Consider the spring – mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is**

**Ans.**(a) As the block oscillates, due to viscous effects, its total energy decreases continuously and so its amplitude decreases continuously. Assuming that the block is initially pulled down and released, its momentum will increases upwards till it reaches the mean position

**Subjective / Numerical integer type**

**31.A point mass m is suspended at the end of massless wire of length L and cross-sectional area A. If Y is the Young’s modulus of elasticity of the material of the wire, obtain the expression for the frequency of the simple harmonic motion along the vertical line. [1978]**

**Ans.**

**32.A mass M attached to a spring oscillates with a period of 2 sec. If the mass is increased by 2kg the period increases by one sec. Find the initial massM assuming that Hooke’s law is obeyed [1979]**

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**33.A pendulum bob of mass 80 mg and carrying a charge of 2 x 10 ^{-8} coulomb is at rest in a horizontal uniform electric field of 20,000 volt/metre. Find the tension in the thread of the pendulum and the angle it makes with the vertical. (Takeg = 9.8ms^{-2}) [1979]**

**Ans.**

**34.Two masses m1 and m _{2} are suspended together by a massless spring of spring constant k. When the masses are in equilibrium, m1 is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of m_{2}.**

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**35.An ideal gas is enclosed in a vertical cylindrical container and supports a freely moving piston of mass M. The piston and the cylinder have equal cross-sectional area A. Atmospheric pressure is P _{Q}, and when the piston is in equilibrium, the volume of the gas is V_{0}. The piston is now displaced slightly down from its equilibrium position. Assuming that the system is completely isolated from its surroundings, show that the piston executes simple harmonic motion and find the frequency of oscillation.[1981-6 marks]**

**Ans.**

**36.A thin fixed ring of radius 1 metre has a positive charge 1 x 10 ^{-5} coulomb uniformly distributed over it. A particle of mass 0.9g and having a negative charge of 1 x10^{6} coulomb is placed on the axis at a distance of 1 cm from the centre of the ring. Show that the motion of the negatively charged particle is approximately simple harmonic. Calculate the time period of oscillations. [1982-5 marks]**

**Ans.**

**37.Two light springs of force constants kj and k _{2} and a block of mass m are in one line AB on a smooth horizontal table such that one ends of each spring is fixed on rigid supports and the other end is free as shown in the figure. The distance CD between the free ends of the springs is 60 cm. If the block moves along AB with a velocity 120 cm/sec in between the springs, calculate the period of oscillation of the block. (k1 =1.8N/m, k_{2} = 3.2N/m, m = 200g) [1985-6 marks]**

**Ans.**The block moves along AB with a velocity of 120 cm/sec, in between the springs. Since AB is a smooth table, the journey from D to C or from C to D is performed with the uniform speed of 120 cm/sec. There is neither an acceleration nor a retardation in this region.

The block moves to right and compresses the spring along DB. The spring offers restoring force and the block comes back to D. Thus half of oscillation is completed in this journey.

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**41.A sphere of radius R is half submerged in liquid of density p. If the sphere is slightly pushed down and release, find the frequency of oscillation. [2004-4 marks]**

**Ans.**

**42.A small body attached to one end of a vertically hanging spring is performing SHM about it’s mean position y _{0} with angular frequency w and amplitude a. If at a height y* from the mean position, the body gets detached from the spring, calculate the value of y* so that the height h attained by the mass is maximum. The body does not interact with the spring during it’s subsequent motion after detachment. (aw^{2} > g). [2005-4 marks]**

**Ans.**

**True / False type**

**43.A simple pendulum with a bob of mass m swings with an angular amplitude of 40°. When its angular displacement is 20°, the tension in the string is greater than mgcos 20°. [1984 – 2marks]**

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**Fill in the blanks type**

**45.When a spring is cut into three equal parts, time period of vibration of a given mass, attached to it becomes………….. times the original time period. [1989 – 2 marks]**

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