Simple harmonic motion Questions and Answers

A block of mass 4 kg is attached to a massless ideal spring and suspended vertically from a ceiling Initially the spring is un stretched when the block is released from rest the Find maximum speed of the block Take spring constan 100 N m and g 10 m s 1 1 m s 3 3 m s ww eeeeeeeee 2 2 m s 4 4 m s

A rod AB of mass m is bent in a shape such that it forms an arc of a circle of radius R The ends A and B of the wire are attached to point O by massless inextensible strings OA and OB If O is the origin X Y are the coordinates of centre of mass of the system and T represents the time period of small oscillations of the system in a plane perpendicular to the plane of OAB then Given angle AOB 60 A X 0 Y 3 3R 2 C T 2n1 R T 3 2 3 3g B X 0 Y D T 2n1 3R 2 RT 3 2 2 I 3 3g A R 1 1 R B

133 A equilibrium towards left and then set free to execute S H M Select correct graph between its velocity v and displacement x 3 4 1 2 f 0 0 0 X X

75 The displacement y of a particle executing periodic motion is given by This expression may be considered to be a result of the superposition of y 4 cos 1 two 3 four sin 1000 t 2 three 4 five independent harmonic motions

3 6T 7 4 32 Find the time period of oscillations of a body placed in a tunnel dug into earth any where in the earth approximately 1600 T 2 100 T 3 2 4 10T 10 sec O sec 2 Th Fx Fx 0 1 n 2 22 15 38 The total for in the figure will be 5 5 1 2 3 33 The angular amplitude of a simple pendulum 39 A body of mass a um tension in the string is spring of spring con change in position 2k 2k

13 A particle is executing SHM on a straight line A and Bare two points at which its velocity is zero It passes through a certain point P AP BP at successive intervals of 0 5 s and 1 5 s with a speed of 3 m s a the maximum speed of particle is 3 2 m s b the maximum speed of particle is 2 m s AP 2 1 c the ratio B P 2 1 AP 1 RP d the ratio IS is 2

executing S H M as shown in the figure 1 FA time graph of a particle 3 y to T12 FA 0 The corresponding force time graph of the particle will be 14 T FA 2 o FA T12 t TRA T12 4

A block of mass m is attached to two unstretched springs of spring constants k and k as shown in figure The block is displaced towards right through a distance x and freeee is released Find the speed of the block as it passes through a distance x 4 from its mean position K K seeeeet mheeeeeee

A needle of a sewing machine moves along a path of amplitude 4 cm with frequency 5 Hz Find its acceleration s after it has crossed the mean Ans 34 2 m s 30 position

m 20 kg is 16 A point mass suspended by by a massless spring of constant 2000 N m The point mass is released when elongation in the spring is 15 cm The equation of displacement of particle as function of time is Take g 10 m s ellee m

35 A system is shown in figure Find the time period for small oscillations of two blocks 1 2 2 dr 3 2 3m k 3m 4k m k 2k wwwwwwww m 3m 2 22K 2 4 2 3m 8k

A plank with a small block on top of it is under going vertical SHM Its period is 2 sec The minimum amplitude at which the block will separate from plank is 10 2 A B 2 10 C 20 2 D 10

acce 15 A particle in S HM has a period Is an amplitude 5 cm Find the acceleration when it at 2 cm from its mean position

02 A tuning fork A produces 4 beats second with another tuning fork of frequency 246 Hz When the prongs of A are filed a little the number of beats heard is 6 per second What is the original frequency of the fork A a 242 Hz b 240 Hz c 250 Hz d 252 Hz

13 The vibration of a string is represented by cos 48nt where x and y are in cm y 5 sin TX 15 and t in second then amplitude of the superposing wave is Acos ut A 2A sinkx 1 1 25 m 3 10 cm In a closed organ pipe the firnt 2 5 cm 4 2 5 cm Z 72 155 Nz 19 19 Which of the fol 1 y Asinkxs 205 5 2 y Asin T a 3 y Acosk vf 4 y Acos 2wt

Two particle execute SHM with same frequency and amplitude along the same straight line They cross each other at a point midway between the mean and the extreme position Find the phase difference between them

21 A load of mass m falls from a height h on to the scale pan hung from a spring as shown in the adjoining figure If the spring constant is k and mass of the scale pan is zero and the mass m does not bounce relative to the pan then the amplitude of vibration is ri a b mg k d mg k mg mg c k k 1 2hk ing mg k 1 2hk mg 1 2hk mg moon E m liga

The amplitude of a SHM reduces to 1 3 in first 20 second then in first 40 second its amplitude becomes bound of 8 1 13 2 3 1 27 1 4 3

The period of oscillation of a body of mass m suspended from a light spring is T When a body of mass m is tied to the first body and the system is made to oscillate the period is 27 Compare the masses m and m Ans 1 3

1 2 Two parallel S H M s represented by x 5sin 4 t 3 cm and x 3sin 4 t 4 m are superposed on a particle Determine the amplitude and epoch of the resultant S H M Ans 7 936 cm 54 23

4 Two bodies M and N of equal masses are suspended from two separate massless springs of force constants k and k respectively If the two bodies oscillate vertically such that their maximum velocities are equal the ratio of the amplitude M to that of N is a k k b k k C k k y Vk

particle is performing simple hormonic motion on straight line with time period T The minimum time taken by the particle to travel from right extreme position to half of its amplitude will be 1 TIT 2 T 4 T

The ratio of maximum velocity to the velocity of a particle performing S H M at a point where potential energy is 25 of total energy is 1 2 3 2 3 2 4 1 3 3 1 2

6 There are two pendulums of length 1 and 2 start vibrating At some instant the two are in mean position in the same phase Calculate after how many vibrations of shorter pendulum the two will be in phase in the mean position 12 4 121 cm 12 100 cm a 11 c 9 b 10 d 8

A cylindrical block of density d stays fully immersed in a large beaker filled with two immiscible liquids of different densities d and d The block is in equilibrium with half of it in liquid 1 and the other half in liquid 2 as shown in the figure If the block is given a displacement downwards and released then neglecting frictional loses A it executes simple harmonic motion B its motion is periodic but not simple harmonic C the frequency of oscillation is independent of the size of the cylinder D the displacement of the centre of the cylinder is symmetric about its equilibrium position 1 2 d d d

d 1 1 A simple harmonic oscillator consists of a particle of mass m and an ideal spring with spring constant k The particle oscillates 15 with a time period T The spring is cut into two equal parts If one part oscillates with the same particle the time period will be T T a 2T b 2T c 5 d

A simple pendulum of length 1 m is allowed to oscillate with amplitude 2 It collides elastically with a wall inclined at 1 to the vertical Its time period will be use g n a 2 3 s c 2 s 1 2 b 4 3 s d none

A10 cm Freqency of oscillation is A 8 B 20 cm B 10 C 30 cm F 2 kaa 2 C 5 kaq xw D 2 f 1 w 271 01 sin 10xt 2 CUM displacement from mean position can be expressed as

Any non harmonic periodic function can be constructed from two or more harmonic functions One such function is F t a sin at a sin 2 t It can be easily checked that the functions tan wot and cot at are periodic with period T w while sec wot and cosec wt are periodic with period T 2 w Thus an sec 1 2 But such functions take values between zero and infinity So these functions cannot be used to represent displacement functions in periodic motions because displacement always takes a finite value in any physical situation 1 tan wt 7 tan ot sec wt 2 sec wt

Given acceleration due to gravity on the surface of earth is y i In the device shown in the figure the block m is displaced down slightly and released it starts oscillating Pulleys are smooth and massless string and springs are also massless The time period of oscillation is T T then the value of will be given K 3N m K 4 N m and m 12 kg T 00 Y wwwww K www Kix m K M

A particle performing linear S H M of period 27 seconds about the mean position O is observed to have a speed of b 3 m s when at a distance b metre from O If the particle is moving away from O at that instant find the time required by the particle to travel a further distance b Ans T 3 s

37 A graph of the square of the velocity v against the square of the acceleration a of a given simple harmonic motion is 1 3 h h 2 a a 4 a CO

55 When the string of a conical pendulum makes an angle of 45 with the vertical its time period is T When the string makes an angle of 60 with the vertical its time period is T2 Then T T2 is a 2 60 Fig 7 307 b 2

A rod of mass M and length L is hinged at its one end and carries a particle of mass m at its lower end A spring of force constant k is installed at distance a from the hinge and another of force constant k at a distance b as shown in the figure If the whole arrangement rests on a smooth horizontal table top the frequency of vibration is A 1 k a k b 15 2T L m M 3 B 1 K k 2 V M m C jk K 1 2 M 4 3 a b 3 6 fum k K 1 21 4 3 k b a m M m nd M m

21 The r m s value of potential difference V shown in the figure is 1 V Dole Vo 0 V 2 I T 2 2 Vo T 2T t 3 4 V 5 2 2

An electric fan is turned off and its angular velocity decreases uniformly from 500 rev min to 20 rev min in 4 seconds a Find the angular acceleration in rev sec b the number of revolutions mad by the motor in the 4 00 seconds interval c How many more seconds are required for the fan to com to rest if the angular acceleration remains constant at the value calculated in part a and b

Two oscillating simple pendulums with time period T 5T and are in phase at a given time They are again in 4 phase after an elapse of time a 4T c 6T b 3T d 5T 2015

If a particle starting from mean position is in with time period T then time taken by part oscillation is complete 1 3 3T 8 T 12 3 8 2 4 5T 12 T

If two SHMs are represented by equations TC 5 3 Y 4 sin 3nt Y 4 sin 3rt 3 cos 3nt then the ratio of their amplitudes is 1 1 2 2 1 1 3 3 1 4 1 3

A body of mass m is situated in a potential field U x U 1 cosax where U and a are constants The time period of small oscillation is m 1 2 Uoa 2 2n 3 2T 4 2 m U a m 2U a 2m U a

O A horizontal plank has a rectangular block placed on it The plank stars oscillating vertically and simple harmonically with an amplitude of 40 cm the block just loses contact with the plank when the later is momentary at rest Then a the period of oscillating is 27 5 s b The block weighs double its weight when the plank is at one of the positions of momentary at rest c the block weight 1 5 times its weight on the plank half way down d The block weights its true weight on the plank when the latter moves fastest

5 A simple pendulum of length 5 m is suspended from the ceiling of a cart Cart is sliding down on a frictionless surface having angle of inclination 60 The time period of the pendulum is 1 2 S 2 S 3 4 S 4 EN 2 S

2018 19 19 For a spring mass system spring having sprin constant 19 7 N m is attached to it What show be the value of mass m approx so that it will g the same period as that of second s pendulum 1 1 kg 2 2 kg 3 3 kg 4 4 kg

In a spring block system shown the block of mass m moves over a smooth horizontal surface and under goes SHM with time period T and amplitude A A constant horizontal force F now begins act on the block Motion of the block will be k m F 1 Periodic but not SHM 2 SHM with time period T 3 SHM with time period T F mA 1 2

y sin oot cos 2ot dy dt cosat 2wsin2wt d y y Oscillatory but S H M not possible dt d y dt w sinot 4w cos2wt wsi

A large number of bullets are fired in all directions with same speed v Maximum area on the ground on which these bullets will spread is 1 TV g 24 3 92 2 TV4 g 4 72V 9 2

Damped harmonic oscillator consists of a block m 2kg a spring k 87 N m a damping force F by Initially it oscillates with an amplitude of 25 cm Because of the damping the amplitude falls to three fourth of this initial value the completion of four oscillations What is the value of b Assume small

The graph between velocity and displacement of a particle executing S H M of angular frequency 1 rad s can be where A is the amplitude of displacement 1 Circle with radius 2 A 2 Circle with radius A 3 Straight line 4 Ellipse

13 A bar of mass M rests in equilibrium on a vertical spring lower end of which is affixed on the ground Now a block of mass m is held on the bar without exerting any force on it and then the block is released How much maximum force the bar will apply on the block during the subsequent motion Acceleration of free fall is g

A particle of mass 4 kg is executing S H M Its displacement is given by the equation Y 8 cos 100t 1 4 cm Its maximum kinetic energy is a 128 J b 64 J c 16 J d 32 J