Simple Harmonic Motion.docx

8/17/2019 Simple Harmonic Motion.docx

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Simple Harmonic Motion (A/L Combined Mathematics)

1. A particle of mass m is attached to one end of an elastic string of natural length a and modulus of elasticity λ. The other end of the

string is attached to a fixed point A on a smooth horizontal plane. Initially the particle is kept at rest at a distance 2a from the point A

and then released. Show that it is in a simple harmonic motion for a timeπ

2 √ma λ .2. A particle P of mass m which lies on a smooth plane is attached to one end of a light spring of natural length l and modulus

elasticity 2mg , whose other end is attached to a fixed point O on the plane. Initially the spring is unstreatched and the particle is

displaced a distance less than l , along the direction of OP and released. Prove that the displacement x of the particle at time t ,

satisfies the euation´ x

!

2 g

l x " #.If the greatest velocity of the particle is √ gl2 , find the amplitude of the motion.3. $ne end of a light elastic string is attached to a fixed point of a ceiling and the other end to a particle which hangs in euili%rium

and causes an extension l in the string. The euili%rium of the particle is distur%ed, at time t " #, %y giving it a velocity & √ gl

vertically downwards.

Prove that

'i( the maximum extension of the string is )l.

'ii( the string %ecomes slack after a time7 π

6

l

g .

'iii( the particle will not hit the ceiling, provided that the natural length of the string exceeds

3 l

2 .

'iv( the time needed to particle to reach its maximum height is (√ 3+ 7π 6 )√ lg when the natural length of the string is greater

than

3 l

2 .

*. A particle of mass m is attached to one end of a light elastic string of natural length l and the other end of the string is attached to a

fixed point $. +hen the particle hangs in euili%rium the extension of the string is

l

3 . ind the modulus of elasticity of the string.


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The particle is held at the point distant

l

2 vertically %elow $, and is released from rest. ind the velocity of the particle when it

first reaches the point A distant l vertically %elow $. -et %e the lowest point reached %y the particle. Show that, for the motion of the

particle from A to , the extension x of the string satisfies the euation´ x

!

3 g

l ( x− l3 ) " #.

Assuming that the solution of the a%ove euation is of the form x " l3 ! cosωt + sinωt , fnd the values o

the constants , and . Hence, fnd the centre and the amplitude o the simple harmonic motion

perormed by the particle rom A to B.

Show that the particle reaches the point B ater a time √ lg {1+ 2π 3√ 3 } rom the instant o release. (20!"/. A and are two points on a smooth horizontal ta%le at a distance 0l apart. A smooth particle P of mass m lies at a point on A in

%etween the points A and . The particle P is attached to the point A %y a light elastic string of natural length )l and modulus ofelasticity * and to the point %y a light elastic string of natural length &l and modulus of elasticity .

If the particle P is at euili%rium at a point 1, show that A1 "

42

11 l .

The particle P is held at the mid2point 3 of A and then is released from rest. +hen the particle P is at a distance x from the point A

along A, o%tain tensions of the two strings

+rite down the euation of motion of the particle P for

40

11 .l # x # *l and show, in the usual notation, that´ x

!

11 λ

6ml ( x−42 l11 ) " #.

Assuming that the solution of the a%ove euation is of the form y " Acosωt

! sinωt

, find the constants A, and .

ind the velocity of the particle P when it is at a point, distant

41

11 l from the point A. ( 2012).

'P.T.$.(

4. A particle P of mass m is attached to one end of a light elastic string of natural length l . The other end of the string is attached to a

fixed point O at a height 4l from a horizontal floor. +hen the particle P hangs in euili%rium, the extension of the string is l . Show that

the modulus of elasticity of the string is mg . The particle P is now held at O and pro5ected vertically downwards with a velocity

√ gl . ind the velocity of the particle when it has fallen a distance l . +rite down the euation of the motion for the particle P ,

when the length of the string is &l ! x, where 6l 7x 7&l, and show that´ x

!

g

l x " #, in the usual notation. Assuming that the


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a%ove euation gives ´ x2

"

g

l ' c

2

2 x2

(, where c '8#( is a constant, find c. Show that the particle P comes to

instantaneous rest when it reaches the floor and that the time taken from O to reach the floor is

1

3 ') √ 3 2) ! &9( √ lg .

( 2011-A/L New)

5. $ne end of a light elastic string of natural length l is attached to a fixed point O of a ceiling. A particle P of mass m is attached to

the other end of the string. +hen the particle P hangs in euili%rium, the extension of the string is l. Show that the modulus of

elasticity of the string is mg .

+hile the particle P hangs in euili%rium, another particle of mass m is held at the point O and released from rest. The particle

moves vertically downwards along the string, under gravity, collides and coalesces with P . Show that the complete particle !

consisting of %oth particles P and will %egin to move with velocity √ gl vertically downwards. $%tain the euation of motion

for the composite particle ! , when the extension of the string is x and show that´ x !

g

2 l 'x2&l( " # , in the usual notation. y

writing y " x 6 &l show that´ y

!

g

2 l y " #. Assuming that the a%ove euation gives´ y2

"

g

2 l 'c

2

2 y2

(,

where c '8#( is a constant, find the value of c. Show that the maximum extension of the string is '& ! √ 3 (l. +hen the composite

particle ! reaches the lowest point, the particle suddenly drops out. :sing the Principle of 1onservation of ;nergy, find the velocity

with which the particle P hits the ceiling.

(2011- A/L Ol")

# . A particle P of mass m is attached to one end of an elastic string of natural length l with the other end of the string %eing attached

to a fixed point O of a ceiling. If λ is the modulus of elasticity of the string, Show that, when the particle P hangs in euili%rium, the

extension a of the string is given %y a "

mgl

λ . The string is now stretched %y a further length $ '8a( such the $P is vertical and

eual to l!a!% and the particle P is released from rest. +hen the length of the string is l!a!x, where 6 a 7 x7 %, write down the

euations of motion of the particle P and show that, ´ x ! ga x "o, in the usual notation. Assuming that the solution of the

a%ove euation is of the form x " A cos √ ga t ! sin √ ga t, find the constants A and %.


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Show that, the particle P performs simple harmonic motion for a time √ ag ( π 2 +α ) where < " sin−1 ( ab ) and the velocity

of the particle P at the time when it leaves simple harmonic motion is

b2−a2

g

a(¿)

√ ¿

upwards. Show also that, the particle P thereafter

moves under gravity and will strike the ceiling with non2zero velocity if %8 a 1+ 2 λ

mg . (2010)

& . A point P moves on the circle x2

+ y2

" a2

with uniform speed a=. If is the foot of the perpendicular from P on >2axis,

show that executes simple harmonic motion with period

2 π

ω .

A light spiral spring of natural length l is fixed at the lower end with its axis vertical. A particle of mass m placed at the upper end can

compress the spring a distance " '?l(, when it is at rest. If the same particle is dropped on the upper end of the spring from a height ' ,

show that the particle will execute a simple harmonic motion with amplitude a" √ d2+2d h , provided l @ a ! d.

In this motion, if the particle remains on the spring for at least an interval of time3 π

2 √ dg , find the maximum value of ( hd )(200)

. Two particles P and of masses m and 3m respectively, hang together in euili%rium at one end of a light elastic string of natural

length l , extending it to a length l*4a , the other end of the string %eing attached to a fixed point O. The particle suddenly falls off. If

the length of the string after a time t is l * x , o%tain the euation

¿

d2

x

dt 2 + g

a ¿ x2a( "#, for x8#.

iven that x"a ! % sin =t !c cos =t, where ω2

"

g

a , is the solution of the a%ove euation, find the values of the constants $

and c.

ind the maximum height reached %y the particle P a%ove the initial position and show that the time taken to reach this height is

√ ag B92


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. $ne end of a light elastic string of natural length l is attached to a fixed point O , while the other end of the string is attached to a

particle of mass m. +hen the particle hangs in euili%rium, the length of the string is

3 l

2 . ind the modulus of elasticity of the

string.

The particle is pulled a distance a vertically down from its euili%rium position, and released from rest there. The displacement at

time t , of the particle, measured downwards from the euili%rium position is x . Show that

d2

xdt

2 +ω2

x"#, where ω2

"2 gl , so long as the string is taut.

'i( In the case when a ,

l

2 find the period and the amplitude of the ensuing motion.

'ii( In the case when a

l

2 *$ , where $ 0 , show that the time taken for the string to first %ecome slack is

√ l

2 g

[π −cos−1

( l

l+2b )].

D>ou may assume that the solution of the euation

d2

x

dt 2 +ω

2

x"# is x"Acos=t ! sin=tE where A and are constants to %e

determined.F (200&)

10. $ne end of a string with natural length l and modulus mg is attached to a fixed point which is distance 2l away from one edge of a

smooth horizontal ta%le. A particle P of mass m is attached to the other end of the string. The particle P is attached to a second

particle G of mass m using a light inelastic string. Initially is kept at the edge of the ta%le so that OP P l and pushed away

gently to start movement of the system from rest. At time t , $P " l*x and, when P is on the ta%le is distance x %elow the surface of

the ta%le. :sing the principle of conservation of mechanical energy or otherwise, show that ´ x2

" ω2

[ l2−(l− x )2 ] ,

where ω2

"

g

2 l . ind the centre and amplitude of the on going simple harmonic motion of the particle P . Show that the

particle P , reaches the edge of the ta%le when t " 9

√ l

2 g and find its speed at the time. (200#)

11. A thin light elastic string of natural length *a and modulus of elasticity 0mg is fixed vertically at its lower end $. A particle p of

mass m is attached to its upper end P, is in euili%rium at a point A vertically a%ove $. Show that $A"

7 a

2 . How another

particle G of the same mass m is attached to P and the system is started moving from rest. Show that the euation of motion of the


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com%ined o%5ect is´ x

"

g

a x , where x is the displacement of the com%ined o%5ect from the point vertically a%ove $ such that

$")a.

-et the lowest point the system reach %e 1. ind the distance $1 and the time taken %y the com%ined o%5ect to travel from A to 1.

+hen the com%ined particle is at 1, the particle G is 5ust removed, show that the euation of motion of P in the resulting motion is

given %y Ý "2 g

a y , where y is the displacement of P from A.

Assuming a solution >"cos=t ! sin=t for this euation , find the values of the constants , and .

ence, show that the time taken %y the particle P to move from 1 to J isπ

3 √ 2ag ,where J is the point vertically a%ove $ suchthat $J"*a. ind also the speed of the particle P when it reaches J. (2014).

+.3.K.P.+anigasekera '.Sc. P.Jip.3.Sc.(

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Simple Harmonic Motion.docx