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PSH – 1

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

New Delhi – 110 018, Ph. : 9312629035, 8527112111

SIMPLE HARMONIC MOTION

C1 Periodic and Oscillatory Motions :

The motion which repeats itself after cetain time of interval is known as periodic motion. For e.g. motion ofa satellite around the planet, uniform circular motion etc. To and fro periodic motions about a fixed point isknown as oscillatory motion. Every oscillatory motion is periodic but the converse is not true.

C2 Period and Frequency :

The smallest interval of time after which the motion is repeated is called its period, denoted by T. Thereciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequencyof the periodic motion denoted by f. The relation between f and T is given by, f = 1/T. The unit of frequencyis s–1 or Hz (hertz).

C3 Displacement in Periodic Motion :

Let the displacement is given by x = f(t), where t is the time. In periodic motion of period T.

f(t) = f(t + T)

Any displacement represented by cosine function or sine function is periodic. Any periodic function can beexpressed as a superposition of sine and cosine functions of different time periods with suitablecoefficients.

C4 Simple Harmonic Motion :

If a particle moves to and fro about a fixed point (equilibrium position) under the application of a force ortorque (called restoring force or torque) which is directly proportional to the displacement (linear orangular), directed towards the fixed point, is called simple harmonic motion.

C5 Displacement, Velocity and Acceleration in SHM

The necessary and sufficient condition for the motion to be simple harmonic (linear) is that the force shouldbe directly proportional to the displacement i.e. F x

F = –kx or kxdt

xdm

2

2

or 0xdt

xd 2

2

2

with m

k2

The solution of the above differential equation representing SHM is given by :

x = A sin (t + )

where A is the amplitude of the motion, is the angular (circular) frequency = 2/T (T = time period) and is phase constant.

For rotational SHM the analogues equation is = 0sin (t + ), where is the angular displacement and

is the maximum angular displacement.

Velocity is given by : )tcos(Adt

dxv or 22

2

2

xAA

x1Av

The maximum and minimum velocity of the particle are vmax

= A and Vmin

= 0.

Acceleration is : xdt

xda 2

2

2

. Thus |amax

| = 2A and |amin

| = 0

Practice Problems :

1. The equation of S.H.M. of a particle is 0kydt

yd2

2

, where k is a positive constant. The time

period of motion is given by

(a)k

2(b)

k

2(c)

2

k(d)

2

k

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2. A particle is executing S.H.M. of period 4 s. Then the time taken by it to move from the extremeposition to half the amplitude is

(a) 1/3 s (b) 2/3 s (c) 3/4 s (d) 4/3 s

3. A particle is vibrating in S.H.M. If its velocities are v1 and v

2 when the displacements from the mean

position are y1 and y

2, respectively, then its time period is

(a) 22

21

22

21

vv

yy2

(b) 2

221

22

21

yy

vv2

(c) 2

221

21

22

yy

vv2

(d) 2

122

22

21

vv

yy2

4. A particle is executing S.H.M. Then the graph of velocity as a function of displacement is

(a) straight line (b) circle (c) ellipse (d) hyperbola

5. A particle is executing S.H.M. Then the graph of acceleration as a function of displacement is

(a) straight line (b) circle (c) ellipse (d) hyperbola

[Answers : (1) a (2) b (3) d (4) c (5) a]

C6 Energy in SHM

(i) Kinetic Energy (K) )xA(m2

1mv

2

1 2222

(ii) Potential Energy (U) 22xm

2

1

(iii) Total energy (E) 22Am

2

1 = constant

Graph for the variation of K, U and E with the position ‘x’ is given by :

Practice Problems :

1. A body executes S.H.M. with an amplitude A. Its energy is half kinetic and half potential when thedisplacement is

(a) A/3 (b) A/2 (c) A/2 (d) A/22

2. When the potential energy of a particle executing simple harmonic motion is one-fourth of itsmaximum value during the oscillation, the displacement of the particle from the equilibriumposition in terms of its amplitude a is

(a) a/4 (b) a/3 (c) a/2 (d) 2a/3

[Answers : (1) c (2) c]

C7 Simple Harmonic Motion and Uniform Circular Motion :

Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in whichthe circular motion occurs.

C8 Method of finding time period of a SHM :

(i) Force method will be used for linear SHM in which F –x F = –kx and hence time period

k

m2T where m is the mass of particle executing SHM and k is a constant quantity..

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(ii) Torque method will be used for angular SHM in which – = –k and hence time period

k

I2T where I is the moment of the inertia and k is a constant quantity..

(iii) Energy method in which total energy of the system executing SHM is constant.

C9 Spring - Block System :

The time period of SHM for a block of mass ‘m’ connected by a spring of spring constant ‘k’ as shown in

figure is given by k

m2T

Practice Problems :

1. The bodies M and N of equal masses are suspended from two separate massless springs of constantsk

1 and k

2, respectively. If the two oscillate vertically such that their maximum velocities are equal,

the ratio of the amplitude of M to that of N is

(a) k1/k

2(b) k

2/k

1(c)

21 k/k (d)12 k/k

2. The vertical extension in a light spring by a weight of 1 kg, in equilibrium, is 9.8 cm. The period ofoscillation of the spring, in seconds, will be

(a)10

2(b)

100

2(c) 20 (d) 200

3. A massless spring, having force constant k, oscillates with a frequency n when a mass m is suspendedfrom it. The spring is cut into two equal halves and a mass 2m is suspended from it. The frequencyof oscillation will now be

(a) n (b) n2 (c) n/2 (d) 2n

[Answers : (1) d (2) a (3) a]

C10 Simple Pendulum :

The time period of a simple pendulum of length L is given by g

L2 .

If length of the pendulum is large, then time period is given by :

R

l

L

lg

l2T where R = Radius of

the earth.

For the pendulum of infinite length i.e., L , 6.84g

R2T minutes

A second pendulum is the simple pendulum having a time period of 2s.

Practice Problems :

1. The time period of a simple pendulum is T. If its length is increased by 2%, the new time periodbecomes

(a) 0.98 T (b) 1.02 T (c) 0.99 T (d) 1.01 T

2. The length of a simple pendulum is increased by 44%. The percentage increase in its time period willbe

(a) 44% (b) 22% (c) 20% (d) 11%

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3. A girl is swinging on a swing in the sitting position. How will the period of swing be affected if shestands up ?

(a) The period will now be shorter

(b) The period will now be longer

(c) The period will remain unchanged

(d) The period may become longer or shorter depending upon the height of the girl.

4. For a simple pendulum the graph between length and time period will be a

(a) hyperbola (b) parabola (c) straight line (d) none of these

[Answers : (1) d (2) c (3) a (4) b]

C11 Physical Pendulum :

The time period of a physical pendulum is given by :Mdg

I2T

I : moment of inertia about the rotational axis passing through point of suspension

d : distance of the center of mass from the point of suspension

M : total mass of the body

Practice Problems :

1. A rod of mass m and length L is suspended from one of the end point of the rod in vertical plane. Thetime period of the rod for small oscillation is given by

(a)g3

L22 (b)

g

L2 (c)

g

L32 (d)

g4

L32

[Answers : (1) a]

C12 Damped Simple Harmonic Motion :

The motion of a simple pendulum, swinging in air, dies out eventually. This is because the air drag and thefriction at the support oppose the motion of the pendulum and dissipate its energy gradually. The pendulumis said to execute damped oscillations. The mechanical energy of this oscillator will decrease with time dueto presence of damping force which is given by F

d = –bv, where v is the velocity of the oscillator and b is a

damping constant, then the displacement of the oscillator is given by x(t) = A e–bt/2m cos ( t + ), where

the angular frequency of the damped oscillator, is given by 2

2

m4

b

m

k . If the damping constant

is small then , where is the angular frequency of the undamped oscillator. The mechanical energy

E of the damped oscillator is given by m/bt2ekA2

1)t(E .

C13 Force Oscillations and Resonance :

If an external force with angular frequency d acts on an oscillating system with natural angular frequency

, the system oscillates with angular frequency d. The amplitude of oscillations is the greatest when

d = , a condition called resonance.

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SINGLE CORRECT CHOICE TYPE

1. Two simple pendulums having lengths 1 m and 16m are both given small displacements in the samedirections at the same instant. They will again be inphase at the mean position after the shorterpendulum has competed n oscillations where n is(a) 1/4 (b) 4(c) 5 (d) 16

2. A flat horizontal board moves up and down inS.H.M. of amplitude A. Then the smallestpermissible value of time period, such that anobject on the board may not lose contact with theboard, is

(a)A

g2 (b)

A

g

(c)g

A (d)

g

A2

3. A simple pendulum has a time period T. Thependulum is completely immersed in a non-viscousliquid whose density is 1/10th of that of thematerial of the bob. The time period of thependulum immersed in the liquid is

(a) T (b) T10

9

(c) T9

10(d)

10

T

4. One end of a long metallic wire of length L is tied tothe ceiling. The other end is tied to a massless springof spring constant K. A mass m hangs freely fromthe free end of the spring. The area of cross-sectionand Young’s modulus of the wire are A and Y,respectively. If the mass is slightly pulled down andreleased, it will oscillate with a time period T equalto

(a)K

m2 (b)

YAK

)KLYA(m2

(c)KL

mYA2 (d)

YA

mL2

5. The displacement y of a particle executing periodicmotion is given by

)t1000sin(t2

1cos4y 2

This expression may be considered to be a result ofthe superposition of

(a) two (b) three(c) four (d) fiveindependent harmonic motions.

6. A uniform cylinder of mass M and cross-sectionalarea A is suspended from a fixed point by a lightspring of force constant k. The cylinder is partiallysubmerged in a liquid of density . If it is given asmall downward push and released, it will oscillatewith time period

(a)gAk

M2

(b)

gAk

M2

(c)k

M2 (d)

gA

M2

7. Two particles execute S.H.M. of the sameamplitude and frequency along the same straightline. They pass one another when going in oppositedirections each time their displacement is half theiramplitude. The phase difference between them is

(a)3

(b)

4

(c)3

2(d)

4

3

8. A mass M is suspended from a massless spring. Anadditional mass m stretches the spring further by adistance x. The combined mass will oscillate on thespring with time period

(a) mg/x)mM(2

(b) x)mM/(mg2

(c) mgx/)mM(2

(d) x)mM/(mg)2/(

9. Electron in an oscilloscope are deflected by twomutually perpendicular oscillating electric fieldssuch that at any time the displacements due to them

are given by x = A cos t, y = A cos

6t . Then

the path of the electron is(a) a straight line having the equation x = y(b) a circle having the equation x2 + y2 = A2

(c) an ellipse having the equation

4/Ayxy3x 222

(d) an ellipse having the equationx2 – xy + y2 = 3A2/4

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10. Two linear simple harmonic motions of equal am-plitudes and frequencies and 2 are impressedon a particle along x and y axes respectively. If theinitial phase difference between them is /2, theresultant path followed by the particle is given by

(a)

2

222

A

x1xy

(b)

2

222

A

x1x2y

(c)

2

222

A

x1x4y

(d)

2

222

A

x1x8y

11. A piece of wood has dimensions a, b and c. Itsrelative density is d. It is floating in water such thatthe side a is vertical. If it is pushed down a littleand then released, the time period of oscillation willbe

(a)g

abc2 (b)

dg

bc2

(c)g

ad2 (d)

g

bcd2

12. A U tube of uniform bore of cross-sectional area ais set up vertically with open ends up. A liquid ofmass m and density d is poured into it. The liquidcolumn will oscillate with a period

(a)g

m2 (b)

dg

ma2

(c)adg

m2 (d)

adg2

m2

13. A body of mass m falls from a height h onto thepan of a spring balance. The masses of the pan andspring are negligible. The force constant of thespring is k. The body sticks to the pan andoscillates simple harmonically. The amplitude ofoscillation is

(a) mg/k

(b) )mg/hk2(1)k/mg(

(c) ))mg/hk2(11)(k/mg(

(d) )1)mg/hk2(11)(k/mg(

14. A simple pendulum is made by attaching a bob of1 kg to a 5 m long copper wire of diameter 0.08 cmand it has a certain period of oscillation. Next a10 kg bob is substituted for the 1 kg bob. If young’smodulus for copper is 12.4 × 1010 N/m2 then thechange in time period is(a) 0 (b) 0.0031 s(c) 0.0062 s (d) none

15. A horizontal spring block system of (force constantk) and mass M executes SHM with amplitude A.When a block is passing through its equilibriumposition an object of mass m is put on it and thetwo move together. The new amplitude is

(a)mM

MA

(b) A

(c)M

mA (d) none

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EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

Comprehension-1

Two identical balls A and B each of mass 0.1 kg areattached to two identical massless springs. Thespring-mass system is constrained to move inside arigid smooth pipe bent in the form of a circle asshown in figure.

The pipe is fixed in a horizontal plane. The centresof the balls can move in a circle of radius 0.06 m.Each spring has a natural length of 0.06 m andforce constant 0.1 N/m. Initially, both the balls aredisplaced by an angle = /6 radian with respect todiameter PQ of the circle and released from rest.

1. The frequency of oscillation of the ball B is

(a) Hz1

(b) Hz

2

(c) Hz3

(d) Hz

4

2. The total energy of the system is

(a) 2 × 10–5 J (b) 22 × 10–5 J

(c) 32 × 10–5 J (d) 42 × 10–5 J

3. The speed of the ball A when A and B are at the twoends of the diameter PQ is

(a) × 10–2 m/s (b) 2 × 10–2 m/s

(c) 3 × 10–2 m/s (d) 4 × 10–2 m/s

Comprehension-2

A mass m is connected to a spring of mass M andoscillates in SHM on a horizontal smooth surface.The force constant of the spring is k andequilibrium length L.

ANSWERS (SINGLE CORRECT CHOICE TYPE)

1. b

2. d

3. c

4. b

5. b

6. a

7. c

8. a

9. c

10. c

11. c

12. d

13. b

14. b

15. a

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4. The KE of the system when mass m has a speed Vis

(a)2V

3

Mm

2

1

(b)

2V4

Mm

2

1

(c)2V

5

Mm

2

1

(d)

2V6

Mm

2

1

5. The time period of oscillation is

(a)K

M3

1m

2

(b) K

M4

1m

2

(c)K

M5

1m

2

(d) K

M6

1m

2

Comprehension-3

A solid cylinder is attached to a horizontalmassless spring so that it can roll without slippingalong a horizontal surface. The spring constant kis 3N/m. If the system is released from rest at a pointin which the spring is stretched by 0.25m. Underthis condition the centre of the mass of the cylinderperforms simply harmonic motion.

6. The translational kinetic energy of the cylinder asit passes the equilibrium position

(a) J16

1(b) J

32

1

(c) J32

3(d) none

7. The rotational kinetic energy of the cylinder as itpasses the equilibrium position is

(a) J16

1(b) J

32

1

(c) J32

3(d) none

8. The friction force acting on the cylinder is

(a) constant (b) zero

(c) varying linearly (d) none

9. The time period of SHM is

(a) independent of mass of the cylinder

(b) independent of spring constant

(c) dependent on amount of extension of thespring

(d) all are incorrect

MATRIX-MATCH TYPE

Matching-1

In column A systems are shown consisting of amassless pulley, a spring of force constantk = 4000 N/m and a block of mass m = 1 kg. If theblock is slightly displaced vertically down from itsequilibrium position and released then the systemperforms SHM. Match the frequency of oscillationof this system given in the column B

Column - A Column - B

(A) (p) 10 Hz

(B) (q) 5 Hz

(C) (r) 20 Hz

(D) (s) none

Matching-2

The frequency of SHM is f.

Column - A Column - B

(A) The frequency of kinetic (p) fenergy

(B) The frequency of (q) 2fpotential energy

(C) The frequency of total (r) f/2energy

(D) The frequency of force (s) not defined

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MULTIPLE CORRECT CHOICE TYPE

1. A block of mass 0.2 kg which slides withoutfriction on a 300 incline is connected to the top ofthe incline by a massless spring of unstretchedlength 23.75 cm and spring constant 80 N/m asshown in figure. Then

(a) The distance moved by the block beforeit stops is 2.5 cm

(b) If the block is pulled slightly pulled downthe incline and released, then the period

of SHM is s10

(c) The time period of SHM depends onangle of inclination of the inclined

(d) none of these

2. A massless spring of force constant 100 N/m is cutinto two halves. The two halves suspendedseparately to support a block of mass M as shownin figure. If the system vibrates then the frequencyof SHM is (10/) Hz. Then

(a) The force constant of each half is200 N/m

(b) The mass of the block is 1 kg

(c) If the springs are connected in series then

frequency of oscillation is Hz5

.

(d) none of these

3. A bullet of mass m strikes a block of mass M(as shown in fig.) The bullet remains embedded inthe block. Then

(a) The velocity of the block immediately

after collision is Mm

mv

(b) The amplitude of oscillation is

)mM(K

mv

(c) The frequency of the oscillatin is

mM

k

2

1

(d) none

4. Consider a particle performing SHM with timeperiod T. Choose the correct statement from thefollowing ?

(a) The time period of oscillation of kineticenergy is T/2

(b) The time period of oscillation ofpotential energy is T/2

(c) The time period of oscillation of speedis T

(d) The time period of oscillation of velocityis T

5. A linear harmonic oscillator of force constant2 × 106 N/m and amplitude 0.01 m has a totalmechanical energy of 160 Joule. Its

(a) minimum potential energy is 60 J

(b) maximum kinetic energy is 100 J

(c) maximum potential energy is 160 J

(d) none

6. A person normally weighing 60 kg stands on aplatform which oscillates up and down simpleharmonically with a frequency 2 Hz and anamplitude 5 cm. If a machine on the platform givesthe person’s weight, then (g = 10 m/s2, 2 = 10),

(a) the maximum reading of the machine willbe 108 kg

(b) the maximum reading of the machine willbe 90 kg

(c) the minimum reading of the machine willbe 12 kg

(d) the minimum reading of the machine willbe zero

Assertion-Reason Type

Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has4 choices (A), (B), (C) and (D) out of which ONLYONE is correct.

(A) Statement-1 is True, Statement-2 is True;Statement-2 is a correct explanationfor Statement-1

(B) Statement-1 is True, Statement-2 is True;Statement-2 is NOT a correctexplanation for Statement-1

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INITIAL STEP EXERCISE

(SUBJECTIVE)

1. A ball is suspended by a thread of length L at thepoint O on the wall PQ which is inclined to thevertical by an angle . The thread with the ball isnow displaced through a small angle away fromthe vertical and also from the wall. If the ball isreleased, find the period of oscillation of thependulum when (a) < , (b) > .

Assume the collision on the wall to be perfectlyelastic.

2. A light rod of length L2 has a small ball of mass m

2

fixed at one end and another ball of mass m1 fixed

on it at a distance L1 from the fixed end. The rod is

supported at its end O and is free to rotate about ahorizontal axis at O. The rod is slightly displacedfrom its equilibrium vertical position and released.Find the frequency of its oscillations about O andthe length of the equivalent simple pendulum.

3. A simple pendulum of length L and mass m has aspring of force constant k connected to it at adistance h below its point of suspension. Find thefrequency of vibrations of the system for smallvalues of amplitude.

4. Two light springs of force constant k1 and k

2 and a

block of mass m are in line AB on a smoothhorizontal table such that one end of each spring isfixed on rigid supports and the other end is free asshown in fig. The distance CD between the free endsof the spring is 60 cm.

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

1. STATEMENT-1 : Every periodic motion isoscillatory.

STATEMENT-2 : Every SHM is oscillatory.

2. STATEMENT-1 : A hollow metal sphere is filledwith water and a small hole is made at its bottom.It is hanging by a long thread and is made tooscillate. The period of oscillation first increasesreaches a maximum value and then it willdecrease as the water flows out from the sphere.

STATEMENT-2 : The time period of the spheredepends on the mass.

3. STATEMENT-1 : The average kinetic energy forone complete oscillation in SHM is one-half of themaximum kinetic energy.

STATEMENT-2 : The average kinetic energy forone complete oscillation in SHM is one-half of themaximum potential energy.

4. STATEMENT-1 : The equation

x = sin t + sin 2t + cos 2t represents SHM.

STATEMENT-2 : The period of the above

function is

2.

5. STATEMENT-1 : The motion of a simplependulum will be SHM even if the amplitude islarge.

STATEMENT-2 : The time period of SHM forsimple pendulum does not depend on amplitudeof oscillation.

(Answers) EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

1. a 2. d 3. b 4. a 5. a 6. a

7. b 8. c 9. d

MATRIX-MATCH TYPE

1. [A-p; B-q; C-r; D-s] 2. [A-q; B-q; C-s; D-p]

MULTIPLE CORRECT CHOICE TYPE

1. a, b 2. a, b, c 3. a, b, c 4. a, b, d 5. a, b, c 6. a, c

ASSERTION-REASON TYPE

1. D 2. C 3. B 4. D 5. D

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If the block moves along AB with a velocity120 cm/sec, in between the springs, calculate theperiod of oscillation of the block. (k

1 = 1.8 N/m,

k2 = 3.2 N/m and m = 200 gm)

Is the motion simple harmonic ?

5. One end of each of two identical springs, each offorce constant 0.5 N/m, are attached on theopposite sides of a wooden block of mass 0.01 kg.The other ends of the springs are connected toseparate rigid supports such that the springs areunstretched and are collinear in a horizontal plane.To the wooden piece is fixed a pointer which touchesa vertical moving plane paper. The wooden piecekept on a smooth horizontal table is now displacedby 0.02 m along the line of springs and released. Ifthe speed of the paper is 0.1 m/s, find the equationof the path traced by the pointer on the paper andthe distance between two consecutive maxima ofthe path.

6. Two masses m1 and m

2 are connected by a spring

of force constant k and are placed on a frictionlesshorizontal surface. Show that if the masses aredisplaced slightly in opposite directions andreleased, the system will execute simple harmonicmotion. Calculate the frequency of oscillation.

7. If 0.3 kg liquid of density 13.6 × 103 is filled in atube of uniform cross-section 1.7 × 10–3m2. One limbof the tube is vertical while the other is inclined toit at 600. If the liquid in the vertical tube is slightlydepressed and released, find the frequency ofoscillation.

8. A block of mass M1 resting on a frictionless

horizontal surface is connected to a spring of springconstant k that is anchored in a nearby wall. A blockof mass M

2 is placed on the top of the first block.

The coefficient of static friction between the twobodies is µ. Assuming that the two bodies movetogether as a unit, find the period of oscillation ofthe system. What is the maximum oscillationamplitude that permits the two bodies to move as aunit ? If the spring is attached to the upper massand the system will perform SHM then find the newtime period ? Is there any change in the maximumoscillation amplitude in this case to permit thesystem to perform SHM as a unit ? Find it.

9. A diatomic molecule has atoms of masses m1

andm

2. The potential energy of the molecule for the

interatomic separation r is given byV(r) = –A + B(r – r

0)2, where r

0 is the equilibrium

separation and A and B are positive constants. Theatoms are compressed towards each other fromtheir equilibrium positions and released. What isthe vibrational frequency of the molecule ?

10. A pulley in the form of a circular disc of mass mand radius r has a groove cut all along itsperimeter. A string whose one end is attached tothe ceiling passes over this disc pulley and its otherend is attached to a spring of spring constant k.The other end of the spring is attached to ceiling asshown in figure. Find the time period of verticaloscillations of the centre of mass assuming that thestring does not slip over the pulley.

FINAL STEP EXERCISE

(SUBJECTIVE)

1. Two identical simple pendulums each of length Lare connected by a weightless spring as shown infig. The force constant of the spring is k.In equilibrium, the pendulums are vertical and thespring is horizontal and undeformed.

Find the time period of small oscillations of thelinked pendulums, when they are deflected from

their equilibrium positions through equaldisplacements in the same vertical plane : (a) in thesame direction, (b) in opposite direction andreleased.

2. A long uniform rod of length L and mass M is freeto rotate in a horizontal plane about a vertical axisthrough its one end. A spring of force constant k isconnected horizontally between one end of the rodand a fixed wall (fig.).

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ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)

1. (a)g

L2 (b)

1sin

g

L2

g

L2. g

LmLm

LmLm

2

1222

211

2211

3. 2

2

mL

khmgL

2

1

4. 2.825 5. .02 sin (100 y + ); 6.28 cm

6.21

21

mm

)mm(K

2

1

7. Hz17

8.k

MM2 21 ,

k

)MM(µg 21 , k

MM2 21 , yes,

kM

g)MM(µM

2

211

9.21

21

mm

)mm(B2

2

1

10.k2

m3T

When the rod is in equilibrium it is parallel to thewall. (a) What is the period of small oscillations thatresult when the rod is rotated slightly andreleased ? (b) What will be the maximum speed ofthe displaced end of the rod, if the amplitude ofmotion is

0 ?

3. A uniform horizontal plank is resting symmetricallyin a horizontal position on two cylindrical drums,which are spinning in opposite directions abouttheir horizontal axes with equal angular velocity.The distance between the axes is 2L and thecoefficient of friction between the plank andcylinder is µ. If the plank is displaced from theequilibrium position along its length and released,show that it performs simple harmonic motion.Calculate also the time period of motion.

4. Two non-viscous, incompressible and immiscibleliquids of densities and 1.5 are poured into thetwo limbs of a circular tube of radius R and smallcross-section kept fixed in a vertical plane as shownin figure Each liquid occupies one-fourth thecircumference of the tube.

(a) Find the angle that the radius vector to theinterface makes with the vertical in equilibriumposition. (b) If the whole liquid is given a smalldisplacement from its equilibrium position, showthat the resulting oscillations are simple harmonic.Find the time period of these oscilations.

5. A mass m is dropped in a tunnel along the diameterof earth from a height h (<< R) above the surface.Find the time period of motion. Is the motion simpleharmonic ?

6. A weightless bar is supported by two springs fromabove and carries a mass m suspended from aspring from its centre as shown in the figure.

Calculate the natural frequency of the oscillationof mass m.

7. A cylinder of radius r and mass m rests on a curvedpath of radius R. Show that the cylinder canoscillate about the bottom position when displacedand left to itself. Find the period of oscillation.Assume that the cylinder rolls without slipping.

8. A hemisphere bowl of radius R has a small pebbleof mass m inside it. The bowl starts rotating with aconstant angular velocity about its axis.

(a) At what angular position from the axisof roation other than = 0, the pebblecan be in stable equilibrium.

(b) Now if pebble is slightly displaced alongthe slope, what will be the frequency ofsmall oscillations.

PSH – 13


New Delhi – 110 018, Ph. : 9312629035, 8527112111

ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)

1. (a)L

g

2

1

(b)

m

k2

L

g

2

1

2. (a)k3

M2 (b)

M

k3L 0

3.µg

L2 4. (a) 11.310 (b) 2.54R

5.g

R4

g

h24

h2R

Rsin 1

6.)kkkkkk4(m

kkk4

2

1

323121

321

7.g

)rR(6

8. (a)

21

R

gcos (b) 22

22

0R

g

psh – 1 simple harmonic motion - einstein classeseinsteinclasses.com/simple_h_m.pdf · expressed...

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