Simple harmonic motion Questions and Answers

A light wooden rod fixed at one end is kept horizontal and a spring is attached perpendicular to the rod at its other end such that it is at its natural length A load of 0 4 kg tied to the free end of the rod causes that end to be depressed by 2 7 cm If this load i set into up and down vibrations the frequency with which it will oscillate is x 3f Hz the X Take g 10m s

49 A body of mass 5 kg hangs from a spring and oscillates with a time period of 27 seconds If the ball is removed the length of the spring will decrease by a g k metres b k g metres d g metres 1994

A solid sphere has density p in the top half and 2p in the bottom half It is kept in a uniform fluid of density 3p with a hinge at the bottom as shown such that it can freely perform oscillations about any horizontal axis through the hinge Radius of sphere is R Find the period of small oscillations if all the points of the sphere are to oscillate in their respective planes parallel to the plane X 3z shown Question Type Single Correct Type 1 2 64R 35g 46R 2 2 45g

Two springs each have spring constant k and relaxed length I They are both stretched a distance I and attached to a mass m and two walls as shown At a given instant the right sprigs constant is somehow magically changed to 3k the relaxed length remains I At a time T 4V k later what is m s position Ignore gravity in this problem t 20 00000000000 m 21 00000000000

A block of 0 5 kg is placed on a horizontal platform The system is making vertical oscillations about a fixed point with a frequency of 0 5 Hz Find the maximum amplitude of oscillation if the block is not to lose contact with the horizontal platform

The potential energy of a particle of mass 0 1 kg moving along the x axis is given by U 5x x 4 J where x is in metres It can be concluded that the particle is acted upon by a constant force the speed of the particle is maximum at x 2 m the particle executes simple harmonic motion the period of oscillation of the particle is 5 s

A block of mass 0 02 kg is connected with spring and is free to oscillate on a horizontal smooth surface as shown The angular frequency of oscillation is 2 rad s 1 The block is pulled by 4 cm from equilibrium position and then pushed towards the spring with a velocity of 8 cm s The amplitude of oscillation is Neglect any damping 1 3 2 cm 2 4 2 cm 3 2 2 cm 4 1 cm 1 3 mm mm mm V 8 cm s mm 4 cm 2 4

A pendulum clock has an iron pendulum 1m long 10 5 C If the temperature rises by 10 C the clock iron A Will lose 8 seconds per day B Will lose 4 32 seconds per day C Will gain 8 seconds per day D Will gain 4 32 seconds per day

11 Two blocks of mass m each are connected to each other by an ideal spring of spring constant k The whole system is moving with speed v towards a perfectly inelastic wall on a smooth surface Plot graph of V V and V with time Consider t 0 as time when B collides with the wall B does not stick to B cm the wall A W Die gath v 2 E 20 m m m k A mmmmm B x 2 1 B D Vem v 2 v 2 E al EK m T x 2 1 t 3 8

3m 3 2 5K 4 2 5K 3 A particle of mass m is placed at lowest point of smooth parabola x2 ay What is the time period of small oscillation 1 2 3 2 a a 2 2 4 2 2a g a 2g

A simple pendulum is being used to determine the vale of gravitational acceleration g a certain place The length of the pendulum is 25 0 cm and a stop watch with resolution measures the time taken for 40 oscillations to be 50 s The accuracy in g is A 4 40 C 3 40 B 2 40 D 5 40

8 A particle is executing SHM along x axis as shown in the figure Consider left to right direction as positive For the small displacement b to b A X 0 b b A 1 Change in acceleration vector is positive 2 Change in velocity vector is negative 3 Change in displacement is negative 4 All of these

Two charges each of magnitude Q are fixed at 2a distance apart and a third charge q of mass m is placed at the mid point of the line joining the two charges all charges being of the same sign If q charge is slightly displaced from its position along the line and released then determine its time period

Ans 13 cm 7 17 A beaker of mass 1 kg contains 2 kg of water and rests on a scale A 2 kg block of aluminum specific gravity 2 70 suspended from a spring scale is submerged in water as shown in figure 7 101 Find the readings of both scales 36 66 N Reading of upper scale 12 34 NI Ans Reading of lower scale

From the displacement time graph of an oscillating particle the maximum velocity of particle is x cm 10 0 10 1 5 cm s 2 3 cm s 3 cm s 4 2 5 cm s A 2 of t s

A block of mass M is attached b the lower end of a vertical spring The spring is hung from ceiling and has force constant value k The mass is released from rest with the spring initially unstretched the maximum extension produced in the length of the spring will be 1 Mg 2k 3 2 mg k 4 4 Mg k 2 Mg k

30 Oscillations 24 A particle is performing SHM with amplitude A and time period T Its position time graph is as shown in figure then its corresponding a t graph is 1 Aw 2 a XA a A A T 2 Aw a aw a

17 Two mutually perpendicular SHMs of equations x a sin cot y b coscot are acting simultaneously on the particle O as shown in the figure The resultant motion of 0 is Ay b coscot Ox a sincot a a simple harmonic motion of amplitude a b b a simple harmonic motion of amplitude a b or an ellipse d a circle coswt q 2 sine 3 2 Sin wo cos wt 2 21 4 a

8 0 cm 22 The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is AIPMT Prelims 2007 1 Zero 2 0 5 T 3 T

A simple pendulum is attached to the roof of a stationary elevator If the time period of oscillation is T when it is stationary what will be time period of oscillation when the elevator falls freely T B AO C 2T 2 D none of these

D 40 cm A thin annular metal disc of inner and outer radii a and b respectively is freely suspended from point on its outer circumference The length of the corresponding equivalent simple pendulum a b a 3b A B 2b a 2b 2 b C 2b 3a b b D

T 8 Which one of the following equations of motion represents simple harmonic motion AIPMT Prelims 2009 1 Acceleration k x a 2 Acceleration k x a 3 Acceleration kx 4 Acceleration K X k x

A block of mass m compresses a spring of stiffness k through a distance 1 2 as shown the figure If the block is not connected with the spring and the impact of the block with the vertical wall is elastic the period of motion of the block is 77777 2 T k m T 4 1 T None of these E k 1 2 1 2

Q14 A particle undergoing SHM has the equation x A sin 2wt p where x represents the displacement of the particle The kinetic energy oscillates with time period a b c 53 2 t F133 d None of these

A ring of radius R is suspended from its circumference and is made to oscillate about a horizontal axis in a vertical plane The length equivalent to a simple pendulum will be Red E RRI GIDI i 1 2R 2 R 3 3R 2

In the given figure an impulse J is given to the block of mass m in the downward direction As a result of the impulse A Both the blocks start moving with J 3m B Both the blocks start moving with in opposite direction m C The centre of mass of the system is moving downwards D The centre of mass of the system is not moving 2m m

A particle performs simple harmonic motion with a period of 2 second The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position 1 is s The value of a to the nearest integer a

A disc of mass m and radius R is attached with a spring of force constant K at its centre as shown in figure At x 0 spring is unstretched The disc is moved to x A and then released There is no slipping between disc and ground Let f be the force of friction on the disc from the ground A 10 f versus t graph will be A f C f 1 C k www O X 0 O X A B f D f

11 Relation between linear momentum p and position x for harmonic oscillator of mass m is as shown in figure The period of oscillation is 1 T 2 T 3 T 4 T 2 b m 2 ma b 2 b 2 b ma 08 P b a X

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 a c Momentum Position Momentum Position freeee b d Momentum 6 Position Momenturn Position

F 1 F 3 r R3 3 R 26 Which of the following quantities is always negative in SHM 2 v j 4 Fxr move on x axis in which c

A particle executing simple harmonic motion has its potential energy 3 times of the kinetic energy taking potential energy at equilibrium to be zero and initially particle starts from equilibrium at same instants during its oscillation Then the instant is are 1 3 T 6 5T 6 2 2T 6 4 All of these

35 A particle of mass m oscillates with simple harmonic motion between points x and x the equilibrium position being O Its potential energy U is plotted It will be as given below in the graph 1 2 O X A AU 3 X X X X 100 sl b Q G J

ent 41 A mass is suspended springs in successive separately by two diffe order then time periods is 2 t and to respectively If it is connected by both the springs as shown in figure then time period is to the correct relation is K 1 t t t 3 1 1 K el man 1 2 t t t 4

Find the angular frequency of particle from given equation Where y displacement and t time 4d y dt 9y 0 Question Type Single Correct Type 1 2 3 9 4 4 9 3 2

block of mass 1 kg is suspended vertically using an ideal spring of spring constant 400 N m When the block is at rest in equilibrium it is given a velocity s in the vertically downward direction Which of these options is are correct g 10 m s 70 The time period of the resulting SHM is 10 The amplitude of the resulting SHM is 7 5 cm The block never reaches any point higher than its initial position The block never achieves a speed greater than its initial speed S

T A particle is performing SHM with its position given as x 2 5 sin t in sec Which of the following is are correct A Equilibrium position is at x 2m B Maximum speed of particle is 5 m s C At t 0 particle is 2 5 m away from mean position moving in negative direction D Att 0 x 4 5 m acceleration of particle is T 4 5 m s t where x in m

In the given figure two elastic rods P and Q are rigidly joined to end supports A small mass m is moving with velocity v between the rods All collisions are assumed to be elastic and the surface is given to be smooth The time period of small mass m will be A area of cross section Y Young s modulus L length of each rod a 2L 2 U 2L c U mL AY mL AY 10 A wire frame in the gh A 2 2Y L P m L b d A Y L Ta 2L U 2L C 2 2mL VAY

Three simple harmonic motion along x axis are simultaneously superimposed on a particle and their independent SHM are given as X A sin ot x A sin wt 7 6 x3 A sin wt 3 Then the resultant amplitude of the oscillation of the particle will be 1 3A 3 1 2 A 2 2A 4 1 3 A

d 50 cm 95 Two pendulums of length 1 m and 16 m start vibrating one behind the other from the same stand At some instant the two are in the mean position in the same phase The time period of shorter pendulum is 7 the minimum time after which the two threads of the pendulum will be one behind the other is no a T 4 c 4 T 3 b T 3 d 4 T OM The dionlocoment of two identical particle executing SHM are represented by eg ons x

A seconds pendulum is suspended from ceiling o a trolley moving horizontally with an acceleration o 4 m s2 Its time period of oscillation is approximately 1 1 60 sec 3 210 sec 2 1 90 sec 4 2 sec

42 A 5 GHz plane wave is propagating in a material characterized by u r 1 r 2 53 and o 0 Find B by assuming that the electric field is given by E V m 2 Points 133 3 rad m 213 4 rad m Mr 1 2 53 o 0 and Er 10cos 10m x 10 t Bz x 57 69 rad m

The potential energy of a particle of mass 0 08 kg moving on the X axis is given as a function of its position xas U x x 8x 2 Then which of the following can represent the position of the particle as a function of time TU Ox t 4 4sin 5t Ox t 4 2 cos t T Ox t 2 2cos 5t Ox r 2 4sin 1 2

10 Two blocks of masses m and m are kept on a smooth horizontal table as shown in figure Block of mass m but not m is fastened to the spring If now both the blocks are pushed to the left so that the spring is compressed a distance d The amplitude of oscillation of block of mass m after the system is released is a d m Vm m c d 2m2 M m oooo m1 b d d d m m m m 2m m m

2 A particle starts oscillating simple harmonically from its equilibrium position Then the ratio of kinetic and potential energy of the particle at time is T 12 Umean 0 T time period a 2 1 b 3 1 a c 4 1 d 1 4

1 A particle moves such that its acceleration is given by a x 2 B Here is a positive constant and x the x coordinate with respect to the origin Time period of oscillations is a 2 B 1 VB c 2 B 2 b 2 1 VB 2 d 2

Three identical springs each of force constant k have been joined to the three identical balls each of mass m as shown in the figure which are in at the three vertices of an equilateral triangle In the shown arrangement each of the spring is in its natural length What all three balls are simultaneously given small displacements of equal magnitude along the directions as shown in the figure the oscillation frequency for the blocks will be then 3 k m 2 V 3k 2m

6 In the figure the block of mass m attached to the spring of stiffness k is in contact with the completely elastic wall and the compression in the spring is e The spring is compressed further by e by displacing the block towards left and is then released If the collision between the block and the wall is completely elastic then the time period of oscillations of the block will be a 5 2 3 Vk EE TC c k mor E b 2 k TC m d wall

7 Vertical displacement of a plank with a body of mass the law m on it is varying according to y sin wt 3 cos ot The minimum value of for which the mass just breaks off the plank and the moment it occurs first time after t 0 are given by y is positive towards vertically upwards g 2 T V2 6 g g T 2 2 3 g a c 3 g 2 T 2 3 g b 2 d 2g 2 3g

C d 50 5 An accurate pendulum clock is mounted on ground floor of a high building How much time will it lose or gain in one day if it is transferred to top storey of a building which is h 200 m higher than the ground floor Radius of earth is 6 4 x 106 m a it will lose 6 2 s c it will gain 5 2 s b it will lose 2 7 s d it will gain 1 6 s