Simple harmonic motion Questions and Answers

A particle moves such that its acceleration a is given by a 8 x where x is the displacement from equilibrium position and a b are constants The period of oscillation is 8 1 2 3 2 a B 2 2 B 4 2 R Va B

A uniform rod of length I is suspended through one end and is set into oscillation with small amplitude under gravity The time period of oscillation is 1 2n 21 3g 3g 3 2 21 2 2 4 2 T 3g

A particle free to move along the x axis has potential energy given by U K 1 exp x for x 0 where K is a positive constant of appropriate dimensions Then 1 At points away from the origin the particle is in unstable equilibrium 2 For any finite non zero value of x there is a force directed away from the origin 1 K 2 3 If its total mechanical energy is it has minimum kinetic energy at the origin 4 For small displacements from x 0 the motion is simple harmonic

A cylinder of mass m radius r is slightly displaced and released Find the frequency of oscillation assuming the cylinder rolls without slipping 1 k voorvoo 3 1 2k 2 3m 1 3m 2 4k 2 4 1 4k 2 3m k 2 m

Question 25 Figure below show a massless pulley a spring of constant K 1000 N m and a mass 1 kg On displacing the mass slightly the frequenc of its vertical oscillation is approximately z x 10 Hz Then find K

The block is released after stretching the spring by a distance A from its natural length The graph which represents the velocity of the block will be 1 3 K vooooooooo m smooth A VA t 2 VA VA X axis ht t 4 t

Consider a particle subjected to a time independent potential V r a Assume that a state of the particle is described by a wave function of the form r t o r x t Show that x t A exp iwt A is constant and r must satisfy the equation A v o r V r o r hw r 2m

In the measurement of the acceleration due to gravity using a simple pendulum the percentage error in the measured value of the length the pendulum is 2 and that in the time period of oscillation is 1 What is the percentage error in the measured value of the accelerati due to gravity A 4 B 5 c 6 D 3

A particle is executing simple harmonic motion about y 0 along y axis Its position at an instant is given by y 7sin nt where y is in metres Its average speed for a time interval t 0 to t 1 5 sec 1 28 m s 2 14 m s 3 7 m s 4 1 7 m s

A particle performs SHM on a straight line with time period T and amplitude A The average speed of the particle between two successive instants when potential and kinetic energy become same is 1 3 A 2A T 2 4 4 2A T 2 2A

A block of mass m is attached to one end of a spring of spring constant k and arrangement is kept at rest on smooth horizontal surface as shown The free end of spring is now pulled with constant speed of vo then maximum energy stored in spring is k voor mee m

Two boys A and B are at a distance 1360 m apart Boy B clapped at the same time boy A released the bob of pendulum from its extreme point Boy A hears the clapping when bob reaches first time at its initial position The time taken by pendulum to oscillate 45 times is Take speed of sound 340 m s

Two particles are executing SHM along the same line with same frequency and amplitude A What is the phase difference between the particles if they cross each other at position 1 RI6 A 2 2 5m 6

K 000000000 000000000 K m K vvvvvvvvv vo In the arrangement shown the spring constant K of each spring is 10 N m and if block of 100 gm is displaced by 2 cm then it will experience the ac celeration of 1 2 m s X3 6 m s NA 2 4 m s zero

Two linear SHM of equal amplitudes A and frequencies o are superimposed along x and y axes respectively If the initial phase difference between them is 2 the resultant path followed by the particle is 1 Elliptical 3 Parabolic 2 Circular 4 Straight line

A particle is performing simple harmonic motion then at what percentage of the amplitude is displacement from mean position to have its kinetic energy exceeds its potential energy by half its total energy 1 50 3 20 2 40 4 25

A thin copper plate of mass m has a shape of a square with a side b and thickness d The plate is suspended on a vertical spring with a force constant k in a uniform horizontal magnetic field B parallel to the plane of the plate Find the period of th small amplitude vertical oscillations of the plate B O O 2 O 21 k Legeeeee m m eb B d k m k m 2e0b B d k m 4 eb B d

53 When an oscillator completes 100 oscillations its amplitude reduced to 1 of initial value What will be its amplitude when it completes 200 oscillations 3 2 a c d 2002 3 8 b 6 9

Potential energy of a particle of mass 2 kg is given as 2x Initially particle is at equilibrium position then time period for small oscillation about stable equilibrium position is Options 1 2 S 2 S 4m 3 S

A cylinder of mass m is allowed to roll on a smooth horizontal table with a spring of spring constant k attached to it so that it executes SHM about the equilibrium position What is the time period I of oscillations Options 1 2 3 T 27 T 27 T 27 T 27 m 2m 3k 3m 2k 3m 4k

A particle of mass 4 kg executes SHM such that its energy of SHM is 36 J Its velocioty at t 0 is 3 m s and displacment from the mear position is 20 cm The equation of its motion is ptions x 20 2 sin 15t 3 x 40 2 sin 20t ST x 40 sin 15t cm 3 5 x 20 2 sin 20t cm 6 cm cm

A simple pendulum in a stationary lift has time period T What would be the effect on the time period when the lift i moves up with uniform velocity ii moves down with uniform velocity iii moves up with uniform acceleration a iv moves down with uniform acceleration a v begins to fall freely under gravity

The displacement of a body executing SHM IS given by x Asin 3rt The first time after t 0 when the speed is maximum is 1 0 33 sec 2 0 16 sec 3 0 25 sec 4 0 5 sec

A car start moving from rest on a horizontal ground such that the position vector of a car with respect to its starting point is given as r bt ct where a and b are positive constants The equation of trajectory of car y f x is C y 6 1 y 3 y b 2 X C y 1 2 2 y x C y 1 2 4 y X

Thich of the following statements is true A B D The rate law for a reaction must depend on the concentrations of all reactants that appear in the stoichiometric equation The rate of a catalyzed reaction must be independent of the concentration of catalyst The rate constant for the reaction is independent of the concentrations of the reacting species There is a single rate determining step in any reaction mechanism

Consider a block suspended with the light rod being horizontal in equilibrium as shown All springs are light Neglect friction everywhere Find the time period for small oscillation of the block Te azu gau zu g 3rch 03 2 CalV HV K Jc uz faur KI T fti g ch i 2 J u U H I J 3i u i i far B ay lllllll 14T 7 www F M 2K 14M 2 5K Correct Answer ll 2M K lllllllM K

Two particles A and B execute SHM of period 5T They start from MP Find the phase difference between them when particle A complete one oscillation T and 1 2 5 4 2 3 5 4 3 5

When the arrangement shown is in equilibrium the spring on the right 62 m is stretched by an amount xo Coefficient of static friction between the blocks is u and the horizontal floor is frictionless Both the blocks have equal mass m and force constants of the spring are 3k and k as shown in the figure Find the maximum amplitude of oscillations of the blocks along the springs that does not allow them to slide on each then Aggoloration of free fall is g 3k 00000000000000000 m m

A block attached to an ideal spring oscillates without friction on a horizontal table A student makes the following measurements on the block k 97 71 m 5 19 A 0 84 Vmax 4 28 Vave 0 76 What is the maximum potential energy of the mass spring system

T 10 2 A simple harmonic equation is given by y 10 sin Calculate the following measurements are in S I units a frequency b period e displacement velocity and acceleration at 1 10s d maximum displacement maximum velocity and maximum acceleration

A 30 g mass is suspended by a vertical spring whose force constant is 15 N m The mass is now immersed in a liquid that provides a damping coefficient of 70 g s for the oscillating system What is the frequency of the damped oscillations

For a particle having simple harmonic motion the displacement is 12 cm at the instant the velocity is 5cm s and the displacement in 5 cm at the instant the velocity is 12 cm s Calculate i amplitude ii frequency and iii time period Hint v a y Ans i 13cm ii r hertz iii 2xs

Displacement x of a particle performing SHM The displacement of a particle is always measured from its mean position The restoring force acting on the particle is given by F kx where we have considered x as the displacement from the equilibrium position or mean position The particle may start its motion from the mean position x 0 extreme position x A or any other position A x A Case I Particle starts from the mean position At t 0 x 0 x A P At t 0 x A At t t X X Displacement x of a particle performing SHM Case II Particle starts from the extreme position At t 0 x A x 0 0 At t t X X At t t3 X X3 Case III General equation of SHM X A At t 0 A x A

The Hamiltonian of a particle having mass m in one dimension is described by p 1 mo x 2ux What is the difference between the energies of the first tw 2m 2 H levels 2 4 mo a hw b ho p c o d ho 24

A 1 8 kg mass is attached to a spring with stiffness 16 N m The damping constant for the system is 2 N sec m If the mass is moved 3 4 m to the up of equilibrium and given an initial upwards velocity of 2 m sec determine a the equation of motion of the mass b give its damping factor amplitude times the exponential c period and frequency of the equation of motion

Problems For problems marked refer to page 522 for a hint clue or answer 6 1 6 2 Oscillation of bead with gravitating masses A bead of mass m slides without friction on a smooth rod along the x axis The rod is equidistant between two spheres of mass M The spheres are located at x 0 y a as shown and attract the bead gravitationally Find the frequency of small oscillations of the bead about the origin nater Oscillation of a particle with two forces A particle of mass m moves in one dimension along the positive x ant force directed toward the origin

mping find out the expressions for amplitude and time period find out the solution for the same Considering the ator subjected to a damping b A mass of 50 gm is attached to the free end of a massless string hung from a rigid support The system oscillates with a frequency of 10 Hz and amplitude of the damped oscillation reduces to one fourth of its un damped value in 2 minutes Calculate the damping coefficient and relaxation time

A particle starts from a point P at a distance from the mean position O and travels towards left as shown in figure If the time period of SHM executed about Ois T and amplitude A then the equation of motion of particle may be written as Yc cu faa zerf g3uz meg fa uz y gra am A B C m A ad gu 2 2 t x Asin Your Answer x Acos 4 T Correct Answer T P A 4 x Asin t 4 Correct Answer

The mass of rod AB is negligible and there are 3 springs attached to it two springs with stiffness coefficients of c1 ans c2 hang the rod at both ends The stiffness coefficient of the third spring is c3 which is tied at the midpoint of the rod and hung with an object P of mass m Find the natural frequency of the object s vibration C1 C3 P C2 B

6 4 37 Two pendulums of length 1 21 m and 1 0 m start vibrating At some instant the two are in the mean position in same phase After how many vibrations of the longer pendulum the two will be in phase 1 10 2 11 3 20 4 21 Stra 3 Hyp 43 A hollow a small and is n the hole of its tim 1 The is e

1 3 2 1 2 21 3g 3g 45 A uniform disc of mass M and radius R is suspended in vertical plane from a point on its periphery Its time period of oscillation is 3 2m 3R 9 2 2R 3g 4 2 2 2 4 2 R 3g 3R 2g

An electric dipole consisting of two equal and opposite point charges q m and dipole length 2 is slightly disturbed angularly from its position of stable equilibrium The minimum time during which its gets aligned with the electric field Only one correct answer A B T FIN C 2 ml qE ml qE ml qE ml

A uniform rod cm is placed on a sharp support O such that AO a 25 cm and OB b 75 cm A spring of force constant k 600 N m is attached to end B as shown in figure To keep the rod horizontal its end A is tied with a thread such that the spring is elongated by y 2 cm The reaction of support on the rod immediately after the thread is burnt is O O O a 20 N 4 N 16 N O b ell B

A particle is executing two differe simple harmonic motions mutually perpendicular of different amplitudes and having phase difference TU of The path of the particle will be 2 A Circular C Parabolic B Straight line D Elliptical

given an angular speed A groove is made on the surface along a radius A particle is gently placed inside the grove at a distance 2 from the centre The speed of the particle w r t table when its distance from the centre is x is O O O w x a 2 W x a w 20 x2 92 w x 4 a x 4

The length of the seconds pendulum is decreased by 0 3 cm when it is shifted from place A to place B If the acceleration due to gravity at place A is 981 cm s the acceleration due to gravity at place B is Take 10 975 cm s A B978 cm s c984 cm s D 981 cm s

7 Two springs A and B are identical except that A is stiffer than B i e k kg If the two springs are stretched by the same force then A a more work is done on B i e WB WA b more work is done on A i e WA WB c work done on A and B are equal d work done depends upon the way in which they are stretched

A particle moves such that its position vector F t coscot i sinot where oo is a constant and t is time Then which of the following statements is true for the velocity v t and acceleration t of the particle 1 v is perpendicular to and is directed towards the origin and a both are parallel to r and a both are perpendicular to i is perpendicular to 7 and is directed 2 3 4

ticle of is attached to 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 a 2 2m k c 2 m k Cumumi momomo A b 2 m d 2 3K 90 m 2k ellelle sellele to me

21 A particle of mass 10 gram executes linear SHM of amplitude 5 cm with a period of 2 second find its potential energy Kinetic energy 1 th second after it has passed through the mean position