Simple harmonic motion Questions and Answers

20 A hollow metal sphere is filled with water through a small hole in it It is hung by a long thread and is made to oscillate Water slowly flows out of the hole at the bottom Select the correct variation of its time period 1 The period will go on increasing till the sphere is empty 2 The period will go on decreasing till the sphere is empty 3 The period will not be affected at all 4 The period will increase first then decrease to initial value till the sphere is empty

A body of mass 36 gm moves with SHM of 22 amplitude A 13 cm and time period T 12 sec At time t 0 the displacement x is 13 cm The shortest time of passage from x 6 5 cm to x 6 5 cm is 1 4 sec 3 6 sec 2 2 sec 4 3 sec

A body of mass 36 gm moves with SHM of amplitude A 13 cm and time period T 12 sec At time t 0 the displacement x is 13 cm The shortest time of passage from x 6 5 cm to x 6 5 cm is 1 4 sec 3 6 sec 2 2 sec 4 3 sec

A mass M is suspended by a spring having a spring constant K In equilibrium position mass M is given a speed u Find further extension in the spring 1 3 ME 2M 2 201 3 u VK u u 2M u Mg K Mg 20000000 M K

1 A spring O where we take is executing equilibrium position x potential energy of spring to be zero The spring is oscillating between Xm and xm position with a mass m attached During motion maximum speed of spring will be NCERT Pg 124 1 2 K m 2 k Xm

7 A particle of mass 10 gm is executing S H M with an amplitude of 0 5 m and angular frequency 10 rad sec The maximum value of force acting on the particle during the course of oscillation is 1 25 N 2 5 N 2 2 5 N 4 0 5 N

Show that the total energy of a particle suspended by a string and vibrating in SHM is directly proportional to spring constant 2 A boat can cross the river of width 900 m on a minimum time of 5 45 minutes If the velocity of the current is 10 8 km hr Then find out the time needed to cross the river in minimun distance 3 A pendulum clock with alpha 6 10 5 C gains 10 sec per day What is the variation in day s temperature 4 when anelectronofcharge e and mass m moves with radius r around the nuclear charge Ze describing the circular orbital Find out

18 A small body of mass 10 gram is making simple harmonic oscillations along a straight line with a time period of 14 seconds and the maximum displacement is 10cm Find the energy of oscillations What is the average kinetic energy during one oscillation 1 32 x 10 2J 16 x 10 J 2 32 x 10 2J 32 x 10 J 3 16 x 10 2J 16 x 10 J 4 16 x 10 2J 32 x 10 J

28 A mass M is performing linear simple harmonic motion Then correct graph for acceleration a and corresponding linear velocity v is v a c a 2 b d a T

6 A block of 0 5kg is placed on a horizontal platform The system is making vertical oscillations about a fixed point with a frequency of 0 5Hz Find the maximum amplitude of oscillation if the block is not lose contact with the horizontal platform 1 0 6542m 3 0 7428m 2 0 9927m 4 0 852m

motions in the 24 A particle is subjected to two simple harmonic same direction having equal amplitudes and equal frequency If the resulting amplitude is equal to the amplitude of individual motions the phase difference between them is T T a 3 2 2 b 3 c T 6 d

5 E 3 a 3 25 A ball of mass 2 kg hanging from a spring oscillates with a time period 21 seconds Ball is removed when it is in equilibrium position then spring shortens by a 10 m d 2 m b 5 m c 20 m 9

9 A particle of mass 0 5kg is executing SHM along a straight line Its path length is 10cm and time period is 8s Calculate its KE PE and TE when its phase angle is 6 radian 1 1 592 x 10 4J 0 562 x10 4J 1 652 x 10 4J 2 2 371 x 10 4J 0 123 x104J 2 154 x 10 4J 3 3 292 x 10 4J 0 462 x104J 5 416 x 10 41 4 2 892 x 10 41 0 964 x10 41 3 856 x 104J

Figure shows a system consisting of a massless pulley and two springs of equal constants k each a block is attached with an ideal string as shown If the block is slightly displaced vertically down from it s equilibrium position and then released the am time period of vertical oscillations is given by 2 then find the value of a k 1 wwwwww k

SHM is x 4sin wt 3sin ot T 3 Here x is in cm and t in sec The amplit of the particle is approximately b 5 cm c 6 cm oscillation a 7 cm d 9 c

A small sphere carrying a charge q is hanging in between two parallel plates by a string of length L Time period of pendulum is To When parallel plates are charged the time period changes to T The ratio T To is equal to 1 p

A spring is compressed by 0 0606 m and is used to launch an object horizontally with a speed of 2 31 m s If the object were attach to the spring at what angular frequency in rad s would it oscillate Number i Units

A particle is performing SHM along x axis with mean position at origin amplitude A and time period T The minimum time A 2 taken by the particle to move from x to x A and return to x 2 OT is given by

Two cyclists Mike and Josh simultaneously started toward each other from two towns d 24 km apart Josh rode at uj 25 km h and Mike at UM 15 km h The moment they start a fly also starts from Josh towards Mike and after reaching Mike immediately returns towards Josh The fly continues back and forth motion between the cyclists till the cyclists meet Air speed of fly is Up 30 km h and the wind blows always towards Mike with a constant velocity u 10 km h Find the total distance s flown by the fly UF uv u 21 km STEUFT Yue Y

A and Care of the same length while B is smaller than A and D is larger than A If A is given a transverse displacement O O 3 Correct Antwer 2 Your Answer 3 Status incorrect D will vibrate with maximum amplitude C will vibrate with maximum amplitude 8 will vibrate with maximum amplitude All the four oscillate with equal amplitude

44 Two springs with negligible masses and force constant of k 200 Nm and k 160 Nm are attached to the block of mass m 10 kg as shown in the figure Initially the block is at rest at the equilibrium position in which both springs are neither stretched nor compressed At time t 0 sharp impulse of 50 N s is given to the block in horizontal direction k elle m K relle frictionless TU a period of oscillations for the mass mis S 6 1 b maximum velocity of the mass m during its oscillation is 10 ms c maximum velocity is 6 m s 5 d amplitude of oscillations is m 6

The velocity of a particle executing SHM is 0 12 ms and 0 04 ms when its displacements from the mean position are 0 01m and 0 03m respectively Calculate the amplitude period and maximum velocity of the particle Ans 0 0316 m 1 57 sec 0 1265 m s

38 The displacement of two identical particles executing SHM are represented by equations x 4sin 10t X1 For what value of same a 16 unit c 4 unit and x2 5cos ot energy of both the particles is b 6 unit d 8 unit

48 A horizontal spring mass system is executing SHM with time period of 4 s At time t 0 it is at mean position Find the minimum time after which its potential energy becomes three times of kinetic energy a 1 s c 1 3 s b 1 2 s d 2 3 s

SCILLATIONS 4 29 A body is executing SHM along line of 4 cm long The velocity of the body at the position of the equilibrium is 6 cm s Calculate i the time period and ii the velocity at a distance 1 cm from the equilibrium position Ans i 2 092s ii 5 196x10 10 m s

15 230 A particle moving about its equilibrium position with equation y ax bt Interpret the condition 1 It will always perform the SHM 2 It can never perform the SHM bx 3 It can perform SHM only when t a bx 4 It can perform SHM only when t a

hinged at a point A to the wall so that it can rotate in all directions The rod is kept in the horizontal position by a vertical inextensible thread of length 6 m fixed at its mid point and other end of the thread is hinged at Sec Sr Super60 All Space for rough work Page 7 Sri Chaitanya IIT Academy 20 06 2021 Sr Super60 All Jee Adv 2018 P1 GTA 11 Q P B as shown in figure The load receives a momentum in the direction perpendicular to the plane of the figure then the time period of small amplitude oscillations of the system is 2 sec then the value of x is X g acceleration due to gravity g m sec neglect viscous forces B

8 A child swinging on a swing in sitting position stands up then the period of the swing will be 1 Increase 2 Decrease 3 Remain same 4 Increase if child is long and decrease if child

downward vertical direction is 30 Total marks 12 a Explain with your own words the Principle of conservation of Energy 4 marks a What is its speed when it reaches the lowest point of its arc 5 marks b What Physics principle did you use to solve a 3 marks 1m 30 F

A torsional pendulum consists of two identical rods each of mass M length joined perpendicular to each other in horizontal plane and is hanged through the wire having torsional constant C The system performs simple harmonic motion 0 0 sin ot such that initially angular displacement of the system is equal to half of amplitude and its angular velocity is and is decreasing then A Time period of pendulum is T 2n BAmplitude of oscillation is X ol2M 3 V C or T Initial phase of SHM is 6 DEnergy stored in wire at t 0 is 7 6 Me2 6C 3 Me 0 36 for Rough Work

mass m density po horizontal cross section area s in equilibrium partially submerged in two containers filled with liquids of densities P and P as shown in figure Find the period of small oscillations of this system about its equilibrium Neglect the changes in the level of liquids in the containers Neglect mass of the strings acceleration due to gravity is g v is volume of each block 1 T 2n Po 2v Po gs P P P Po 2 T 27 v P P T 2T 2v P P gs Po V Po 5

A tunnel is drilled along the diameter of the earth A particle executes S H M in this tunnel and another particle performs uniform circular motion along the orbit near earth s surface The two particles start their journey simultaneously from one end of the tunnel A They will reach the other end of tunnel simultaneously B when the particle reaches the other end of the tunnel the first particle will cover one fourth of the circle C when the particle reaches the other end of the tunnel the first particle will cover complete circle D when the particle reaches the other end of the tunnel the first particle will cover three fourth of the circle

37 The shortest distance travelled by a particle executing S H M from 3 2 its mean position in 2 seconds is of its amplitude Find its Ans 0 21 s period 38 At what displacement the velocity of the body executing S H M is half that at equilibrium position

d 180 290 A boy throws a ball upwards with velocity vo 20m s The L e 90 2 wind imparts a horizontal acceleration of 4 m s to the left The angle 0 at which the ball must be thrown so that the ball returns to the boy s hand is g 10 m s a tan 1 2 c tan 2 291 There are two values of t b tan 0 2 0 4 1 d tan Rotate

Case 2 Equilibrium of a current carrying conductor When a finite length current carrying wire is kept parallel to another infinite length current carrying wire it can suspend freely in air as shown below Movable Fixed mg X 40 Magnetic Effect of Current 12 h Y Downloaded from ToraLabs proof In both the situations for equilium of XY it s downward weight upward magnetic force i e Mo 21 12 1 4 T h Note In the first case if wire XY is SHM and it s time period is give by T 27 h Fixed g Movable lac from its equilibrium position it executes

A heavy block is attached to the ceiling by a spring that has a force constant k A conducting rod is attached to block The combined mass of the block and the rod is m The rod can slide without friction along two vertical parallel rails which are a distance L apart A capacitor of known capacitance C is attached to the rails by the wires The entire system is placed in a uniform magnetic field B Find the time period T of the vertical oscillations of the block Neglect the electrical resistance of the rod and all wires rail x X x X rail

A simple pendulum of length L is constructed from a point object of mass m suspended by a massless string attached to a fixed pivot point A small peg is placed a distance 2L 3 directly below the fixed pivot point so that the pendulum would swing as shown in the figure below The mass is displaced 5 degrees from the vertical and released How long does it take to T return to its starting position T The value of L fixed pivot point small peg point object

c 384 Hz d 500 Hz 29 A particle executes SHM with an amplitude of 2 cm When the particle is at 1 cm from the mean position the magnitude of its velocity is equal to that of its acceleration Then its time period in seconds is b 2n 3 d 2n 3 2 c 3 2 50 A body of mass m thrown horizontally with velocity

3 The maximum velocity of a particle executing simple harmonic motion is v If the amplitude is doubled and the time period of oscillation decreased to 1 3 of its original value the maximum velocity becomes a 18v b 12v c 6v d 3v

When displaced and released the 2 kg mass in figure oscillates on the frictionless horizontal surface with period 6 seconds If a small mass is placed on the 2 kg block and the coefficient of static friction between the small mass and the 2 kg block is 0 1 then the maximum amplitude of oscillation before the small mass slips is A 10 m Assume the period is unaffected by adding the small mass Then value of A is k 100000 m 2kg

In simple harmonic motion Which one of the following quantities has constant ratio with acceleration A Time B Displacement C Velocity Mass Medium

y A 55 P t 0 X Fig 14 10 P t 0 B T 30 s Example 14 4 Fig 14 10 depicts two circular motions The radius of the circle the period of revolution the initial position and the sense of revolution are indicated on the figures Obtain the simple harmonic motions of the x projection of the radius vector of the rotating particle P in each case

In the given figure a mass M is attached to a horizontal spring which is fixed on one side to a rigid support The spring constant of the spring is k The mass oscillates on frictionless surface with time period T and amplitude A When the mass is in equilibrium position as shown in the figure another mass m is gently fixed upon it The new amplitude of oscillations will be 4 M oooooooo k

154 1 The displacements of the particle from the extreme position when its kinetic energy is th of 3 the maximum value and th of the maximum value are XA XA and X respectively The ratio is XB 1 3 1 2 1 2 3 1 3 4 2 3 1 155 15

a block of mass 200 g falls from a height of 9 8 cm on the pan of a spring balance as shown the mass of the pan and spring is negligible the block gets stuck to the pan and starts oscillting simple harmonically in the vertical direction the energy of its oscillation is O 0 1931 O 193 O 19 3j 00103

Motion 11 A simple pendulum has a length of 13 m and mass 25 kg The pendulum bob is pulled 0 5 m to one side from its rest position and releas Calculate the maximum acceleration experienced by the bob in m s 2

Problem 4 7 When the masses of the coupled pendulums of Figure 4 1 are no longer equal the equations of motion become 100 and m x m g l x s x y m y m g l y x x y Show that we may choose the normal coordinates Coupled Oscillations mix m y m m with a normal mode frequency w g l and Y x y with a normal mode frequency X

39 A weightless spring of length 60 cm and force constant 100 N m is kept straight and unstretched on a smooth horizontal table and its ends are rigidly fixed A mass of 0 25 kg is attached at the middle of the spring and is slightly displaced along the length The time period of the oscillation of the mass is a c k a 20 HIS 5 S S T b S 10 d 7 200 S 44 A sc den app fore to at sp sp nc pl a c

Q5 A rectangular block of mass m and area of cross section a floats in a liquid of density p If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period T Then T is proportional to O A 1 a O B m O C 1 m O D P Physics son Remainin

The bob of a pendulum has mass of 40 mg and charge q 10 C is given to it length of pendulum is 90 cm The poin of suspension support also has a point charge of 10 C What is the time period of oscillation of pendulum 9 Support 1 q mg 0 6T S 0 6 s Vo 6 s