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Unformatted text preview: Chapter 8 Simple Harmonic Motion 8 SIMPLE
HARMONIC
MOTION
Objectives
After studying this chapter you should
• be able to model oscillations; • be able to derive laws to describe oscillations; • be able to use Hooke's Law; • understand simple harmonic motion. 8.0 Introduction One of the most common uses of oscillations has been in timekeeping purposes. In many modern clocks quartz is used for this
purpose. However traditional clocks have made use of the
pendulum. In this next section you will investigate how the
motion of a pendulum depends on its physical characteristics.
The key feature of the motion is the time taken for one complete
oscillation or swing of the pendulum. i.e. when the pendulum is
again travelling in the same direction as the initial motion. The
time taken for one complete oscillation is called the period. 8.1 Pendulum experiments Activity 1 Your intuitive ideas To begin your investigation you will need to set up a simple
pendulum as shown in the diagram. You will need to be able to
• vary the length of the string; • vary the mass on the end of the string; • record the time taken for a particular number of oscillations. Once you are familiar with the apparatus try to decide which of
the factors listed at the beginning of the next page affect the
period. Do this without using the apparatus, but giving the
answers that you intuitively expect. The mass is attached by a string
to the support, to form a simple
pendulum. 191 Chapter 8 Simple Harmonic Motion (a) The length of the string (b) The mass of the object on the end of the string. (c) The initial starting position of the mass. Now try simple experiments to verify or disprove your intuitive
ideas, using a table to record your results. You are now in a position to start analysing the data obtained,
using some of the basic mathematical concepts in pure
mathematics. Activity 2 Analysis of results You will probably have observed already that as you shortened the
string the period decreased. Now you can begin to investigate
further the relationship between the length of the string and the
period.
•
• period Plot a graph for your results, showing period against length of
string. length of string Describe as fully as possible how the period varies with the
length of the string. You may know from your knowledge of pure mathematics how a
straight line can be obtained from your results using a loglog plot.
• Plot a graph of log (period) against log (length of string), and
draw a line of best fit. • Find the equation of the line you obtain and hence find the
relationship between the period and the length of the string. log
(period) log (length) Loglog graph 8.2 Pendulum theory You will have observed from Section 8.1 that the period of the
motion of a simple pendulum is approximately proportional to the
square root of the length of the string. In this section you are
presented with a theoretical approach to the problem
The path of the mass is clearly an arc of a circle and so the results
from Chapter 7, Circular Motion, will be of use here. It is
convenient to use the unit vectors, er and eθ , directed outwards
along the radius and along the tangent respectively. The
acceleration, a, of an object in circular motion is now given by eθ ˙
˙˙
a = − r θ 2 er + r θ eθ
where 192 2
˙˙
˙ d θ and θ = d θ .
θ=
dt
dt 2 er The unit vectors for
circular motion Chapter 8 Simple Harmonic Motion Forces acting on the pendulum
As in all mechanics problems, the first step you must take is to
identify the forces acting. In this case there is the tension in the
string and the force of gravity. There will, of course, also be air
resistance, but you should assume that this is negligible in this
case. The forces acting and their resultant are summarised in
the table below.
Force Component form T − Ter mg m g cos θ er − mg sin θ eθ Resultant ( m g cos θ − T )er − mg sin θ eθ θ
T mg Now it is possible to apply Newton's second law, using the
expression for the acceleration of an object in circular motion F = m a,
giving ( m g cos θ − T )er − mg sin θ eθ ( ˙
˙˙
= m − l θ 2 er + l θ eθ ) where r = l, the length of the string.
Equating the coefficients of er and eθ in this equation leads to
˙
T − mg cos θ = mlθ 2
and ˙˙
− mg sin θ = ml θ . From equation (2) (1)
(2) g sin θ
˙
θ˙ = −
.
l If the size of θ is small, then sin θ can be approximated by θ ,
so that gθ
˙
θ˙ = −
l (3) 193 Chapter 8 Simple Harmonic Motion Activity 3 Solving the equation Verify that g θ = A cos
t + α
l is a solution of equation (3), where α is an arbitrary constant. Interpreting the solution
Each part of the solution
g θ = A cos
t + α
l describes some aspect of the motion of the pendulum.
• • The variable, A, is known as the amplitude of the oscillation.
In this case the value of A is equal to the greatest angle that
the string makes with the vertical.
g
determines how long it takes for one complete
l
oscillation. g
l
t = 2 π or t = 2 π
l
g When then the pendulum has completed its first oscillation.
This time is known as the period of the motion.
In general, if you have motion that can be described by
x = A cos(ω t + α )
then the period, P, is given by P= 2π
.
ω It is sometimes also useful to talk about the frequency of an
oscillation. This is defined as the number of oscillations per
second.
• The constant α is called the angle of phase, or simply the
phase, and its value depends on the way in which the
pendulum is set in motion. If it is released from rest the
angle of phase will be zero, but if it is flicked in some way,
the angle of phase will have a nonzero value. 194 2π A –A l
g Chapter 8 Simple Harmonic Motion Exercise 8A
1. A D.I.Y magazine claims that a clock that is fast
(i.e. gaining time) can be slowed down by
sticking a small lump of Blutac to the back of
the pendulum. Comment on this procedure.
2. A clock manufacturer wishes to produce a clock,
operated by a pendulum. It has been decided
that a pendulum of length 15 cm will fit well into
an available clock casing. Find the period of this
pendulum.
3. The result obtained for the simple pendulum
used the fact that sin θ i s approximately equal to
θ f or small θ . For what range of values of θ i s
this a good approximation? How does this affect
the pendulum physically? 8.3 4. A clock regulated by a pendulum gains 10
minutes every day. How should the pendulum be
altered to correct the timekeeping of the clock?
5. Two identical simple pendulums are set into
motion. One is released from rest and the other
with a push, both from the same initial position.
How do the amplitude and period of the
subsequent motions compare?
6. The pendulums in Question 5 have strings of
length 20 cm and masses of 20 grams. Find
equations for their displacement from the
vertical if they were initially at an angle of 5° t o
the vertical and the one that was pushed was
given an initial velocity of 0.5ms −1 . Energy consideration An alternative approach is to use energy consideration.
As the simple pendulum moves there is an interchange of kinetic
and potential energy. At the extremeties of the swing there is
zero kinetic energy and the maximum potential energy. At the
lowest point of the swing the bob has its maximum kinetic
energy and its minimum potential energy. Activity 4 Energy For the simple pendulum shown opposite, find an expression for
the height of the pendulum bob in terms of the angle θ . You
may assume that you are measuring from the lowest point. θ
l Explain why the potential energy of the bob is given by
m gl (1 − cos θ ). v 1
mv 2 .
2
As no energy is lost from the system, the sum of potential and
kinetic energies will always be constant, giving
The kinetic energy of the pendulum bob is given by T = mgl (1 − cos θ ) + 1
mv 2
2 where T is the total energy of the system. The value of T can be
found by considering the way in which the pendulum is set into
motion.
195 Chapter 8 Simple Harmonic Motion Finding the speed
Solving the equation for v gives 2T
− 2 gl (1 − cos θ ) .
m v= This allows you to calculate the speed at any position of the
pendulum.
You can also find an expression for θ from the equation for v. Activity 5 Finding θ Explain why the speed of the pendulum bob is given by
v=l dθ
.
dt (You may need to refer to the Chapter 7, Circular Motion if you
find this difficult.)
Substitute this into the equation for v and show that
dθ
=
dt 2T 2 g
−
(1 − cos θ ) .
ml 2
l Simplifying the equation
If you assume that the oscillations of the pendulum are small,
then you can use an approximation for cos θ . Activity 6 Small θ approximation Use the approximation
cos θ ≈ 1 − to show that θ2
2 dθ
can be written in the form
dt
dθ
= ω a 2 − θ 2 , where a is a constant.
dt 196 Chapter 8 Simple Harmonic Motion Integrating the equation
You can solve this equation by separating the variables to give ∫ 1
a − θ2
2 d θ = ∫ ω dt . The LHS is simply a standard integral that you can find in your
tables book and the RHS is the integral of a constant. Activity 7 Integrating Show that θ = α sin(ω t + c )
and explain why this could be written as θ = α cos(ω t + α )
where α and ω are defined as above and c is an unknown
π
constant given by α = c + .
2
This result is identical to that obtained earlier. Exercise 8B
1. A pendulum consists of a string of length
30 cm and a bob of mass 50 grams. It is released
from rest at an angle of 10 ° to the vertical. Draw
graphs to show how its potential and kinetic
energy varies with θ . F ind the maximum speed
of the bob. 8.4 2. Find expressions for the maximum speed that can
be reached by a pendulum if it is set in motion at
an angle θ ° t o the vertical if
(a) it is at rest;
(b) it has an initial speed u . Modelling oscillations In this section you will investigate other quantities which
change with time, can be modelled as oscillations and can be
described by an equation in the form x = a cos(ω t + α ).
There are many other quantities that involve motion that can be
described using this equation.
Some examples are the heights of tides, the motion of the needle
in a sewing machine and the motion of the pistons in a car's
engine. In some cases the motion fits exactly the form given
above, but in others it is a good approximation. 197 Chapter 8 Simple Harmonic Motion Activity 8
Find other examples of motion that can be modelled using the
equation x = a cos(ω t + α ). Fitting the equation to data
One example that could be modelled as an oscillation using the
equation is the range of a tide (i.e. the difference between high
and low tides). The table shows this range for a threeweek
period.
Day Range Day Range Day Range 1 4.3 8 5.7 15 3.7 2 3.8 9 6.3 16 3.0 3 3.3 10 6.6 17 2.8 4 3.0 11 6.5 18 3.1 5 3.2 12 6.0 19 3.7 6 4.0 13 5.5 20 4.3 7 4.9 14 4.6 21 4.7 The graph below shows range against day.
Tidal range (metres)
7 6 5 3.8
4 3 2 13
1 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Day 198 Chapter 8 Simple Harmonic Motion This graph is clearly one that could be modelled fairly well as an
oscillation using x = a cos(ω t + α ). One difference you will observe
between this graph and those for the simple pendulum is that this
one is not symmetrical about the time axis. The curve has, in fact,
been translated upwards, so the range will be described by an
equation of the form
R = a cos(ω t + α ) + b.
The difference between the maximum and minimum range is
approximately 3.8 m. The amplitude of the oscillation will be half
this value, 1.9 m. So the value of a in the equation will be 1.9.
As the graph is symmetrical about the line r = 4. 7 the value of b
will be 4.7.
It is also possible from the graph to see that the period is 13 days.
From Section 8.2 you will recall that the period is given by
P= ω= so that 2π
ω
2π
.
P In this case, ω = 2π
, which is in radians.
13 This leaves the value of α to be determined. The equation is now
2π t
R = 1.9 cos
+ α + 4. 7. 13 When t = 1, R = 4.3, so using these values
2π
4.3 = 1.9 cos
+ α + 4. 7 13 ⇒ 2π
1.9 cos
+ α = −0. 4 13 ⇒ 2π
0. 4
+ α = −
cos 13 1.9 ⇒
⇒ 2π
+ α = 1. 78
13
2π
α = 1. 78 −
13 α ≈ 1.30
which gives
2π t
R = 1.9 cos
+ 1.3 + 4. 7 13 (Note that in applying this result you must use radians.)
199 Chapter 8 Simple Harmonic Motion Activity 9
Draw your own graph of the data on tidal ranges.
Superimpose on it the graph of the model developed above.
Compare and comment. Exercise 8C
1. In the UK 240 volt alternating current with a
frequency of 50 Hz is utilised. Describe the
voltage at an instant of time mathematically.
2. The tip of the needle of a sewing machine moves
up and down 2 cm. The maximum speed that it
reaches is 4 ms −1 . F ind an equation to describe
its motion.
3. Black and Decker BD538SE jigsaws operate at
between 800 and 3200 strokes per minute, the tip
of the blade moving 17 mm from the top to the
bottom of the stroke. Find the range of maximum
speeds for the blade.
4. A person's blood pressure varies between a
maximum (systolic) and a minimum (diastolic)
pressures. For an average person these pressures
are 120 mb and 70 mb respectively. Given that
blood pressure can be modelled as an oscillation,
find a mathematical model to describe the
changes in blood pressure. It takes 1.05 seconds
for the blood pressure to complete one
oscillation. 8.5 5. The motion of the fore and hind wings of a
locust can be modelled approximately using the
ideas of oscillations. The motion of the fore
wings is modelled by
F = 1.5 + 0.5sin (1.05t − 0.005) where F i s the angle between the fore wing and
the vertical. Find the period and the amplitude
of the motion.
Each hind wing initially makes an angle of 1.5°
to the vertical. It then oscillates with period
0.06 s and amplitude 1.5° . Form a model of the
form h = H + a sin ( kt )
for the motion of one of the hind wings. Springs and oscillations In Section 6.5, Hooke's Law was used as the model that is
universally accepted for describing the relationship between the
tension and extension of a spring. Hooke's Law states that T = ke
where T is the tension, k the spring stiffness constant and e the
extension of the spring. If the force is measured in Newtons and
the extension in metres, then k will have units Nm −1.
To begin your investigations of the oscillations of a mass/spring
system you will need to set up the apparatus as shown in the
diagram opposite. 200 Masses can be attached to
the spring suspended from
the stand Chapter 8 Simple Harmonic Motion The equipment you will need is
• a stand to support your springs
• a variety of masses
• 2 or 3 identical springs
• a ruler
• a stopwatch. Activity 10 Intuitive ideas
If you pull down the mass a little and then release it, it will
oscillate, up and down. Once you are familiar with the
apparatus, try to decide how the factors listed below affect the
period. Do this without using the apparatus, giving the answers
that you intuitively expect.
(a) The mass attached to the spring. (b) The stiffness of the spring. (c) The initial displacement of the mass. Also consider some simple experiments to verify or disprove
your intuitive ideas. Theoretical analysis
To begin your theoretical analysis you need to identify the
resultant force.
.
Explain why the resultant force is ( m g − T )i Activity 11
Use Hooke's Law to express T in terms of k, l and x, where k is
the stiffness of the spring, l the natural length of the spring and x
the length of the spring. So the resultant force on the mass is
i ,
( m g − k x + kl )i
d2x
and the acceleration of the mass will be given by 2 i
.
dt x
T m mg 201 Chapter 8 Simple Harmonic Motion Activity 12
Use Newton's second law to obtain the equation d2x k
kl
+ x = g+
2
dt
m
m
and show that
k mg
x = a cos
t + α +
+l
k
m satisfies this equation. Why is the period of these oscillations given by P = 2π m
?
k Work done
lo Consider a spring of stiffness k and natural length l0 . If a force,
F, is applied to cause the spring to stretch, then this force must
increase as the spring extends. So the work done in stretching a
spring is evaluated using e
F ∫ e ∫ e Work done = Fd x
0 = kx dx
0 e
1
= k x2 2
0 1
= ke 2 .
2 12
ke . When the
2
spring is released the energy is converted into either kinetic or
potential energy. So the energy stored in a spring is given by Simple harmonic motion
If the equation describing the motion of an object is of the form
x = a cos(ω t + α ) + c
then that type of motion is described as simple harmonic
motion (SHM).
202 Work done in stretching
a spring Chapter 8 Simple Harmonic Motion This equation is deduced from the differential equation d2x
+ ω2x = b
dt 2
Both the simple pendulum and the mass/spring system are
examples of SHM. Exercise 8D
1. A clock manufacturer uses a spring of stiffness
−1 40 Nm . I t is required that the spring should
complete four oscillations every second. What
size mass should be attached to the spring?
What initial displacement would be required? 7. The spring in a pinball machine is pulled back
with a plunger and then released to fire the balls
forward. Assume that the spring and the ball
move in a horizontal plane. 2. A 250 gram mass is attached to a spring of (b) the period of its motion;
(c) the amplitude of its motion.
3. A baby bouncer is designed for a baby of
average mass 18 kg. The length of the elastic
string cannot exceed 80 cm, in order to ensure
flexibility of use. Ideally the bouncer should
vibrate at a frequency of 0.25 Hz. Determine the
stiffness constant of the elastic string. 45˚ 5 cm cm 5. Two identical springs are attached to the 2 kg
mass that rests on a smooth surface as shown. from an elastic string of stiffness 80 Nm −1 . T he
diagram below shows the initial dimensions of
the catapult, before the elastic is stretched. The
stone is placed in the catapult and pulled back
5 cm. 10 4. Two identical springs are used to support
identical masses. One is pulled down 3 cm from
its equilibrium position. The other is pulled
down 2 cm from its equilibrium position. How
do the amplitude and period of the resulting
motion compare? 8. A catapult is arranged horizontally. It is made cm (a) an expression for its position; The spring has stiffness 600 Nm −1 c ompressed by
5 cm to fire the ball. The mass of the ball is 50
grams. Find its speed when it leaves contact
with the plunger. 10 natural length 40 cm and stiffness 200 Nm −1 .
The mass is pulled down 3 cm below its
equilibrium position and released. Find 45˚ Find the work done in pulling the catapult back
and the speed of the stone when it leaves the
catapult. The stone has a mass of 25 grams.
60 cm
The springs have stiffness 30 Nm −1 a nd natural
lengths 25 cm. The mass is displaced 2 cm to
the left and then released. Find the period and
amplitude of the resulting motion.
6. A mass/spring system oscillates with period
0.07 s on earth. How would its period compare
if it were moved to the moon? Find the speed of the stone if the catapult is
arranged to fire the stone vertically rather than
horizontally.
9. Turbulence causes an aeroplane to experience an
up and down motion that is approximately simple
harmonic motion. The frequency of the motion is
0.4 Hz and the amplitude of the motion is 1 m.
Find the maximum acceleration of the aeroplane. 203 Chapter 8 Simple Harmonic Motion 10. A buoy of mass 20 kg and height 2 m is floating
in the sea. The buoy experiences an upward
force of 400 d , where d i s the depth of the bottom
of the buoy below the surface. 11. A metal strip is clamped at one end. Its tip
vibrates with frequency 40 Hz and amplitude
8 mm. Find the maximum values of the
magnitude of the acceleration and velocity. (a) Find the equilibrium position of the buoy.
(b) Find the period and amplitude of the motion
of the buoy if it is pushed down 0.3 m from
its equilibrium position. 8.6 Miscellaneous Exercises
1. A seconds pendulum is such that it takes one
second for the pendulum to swing from one end
of its path to the other end, i.e. each half of the
oscillation takes 1 second.
(a) Find the length of the seconds pendulum.
(b) A seconds pendulum is found to gain one
minute per day. Find the necessary change in
length of the pendulum if the pendulum is to
be made accurate.
2. A simple pendulum oscillates with period
2 t s econds. By what percentage should the
pendulum length be shortened so that it has a
period of t s econds?
3. A particle is moving in simple harmonic motion
and has a speed of 4 ms −1 w hen it is 1 m from the
centre of the oscillation. If the amplitude is 3 m,
find the period of the oscillation.
4. A light elastic string with a natural length of
2.5 m and modulus 15 N, is stretched between
two points, P and Q, which are 3 m apart on a
smooth horizontal table. A particle of mass 3 kg
is attached to the midpoint of the string. The
particle is pulled 8 cm towards Q, and then
released. Show that the particle moves with
simple harmonic motion and find the speed of the
particle when it is 155 cm from P.
5. A particle describes simple harmonic motion
about a point O as centre and the amplitude of
the motion is a m etres. Given that the period of
π
the motion is
s econds and that the maximum
4
speed of the particle is 16 ms −1 , f ind
(a) the speed of the particle at a point B, a
1
distance a f rom O;
2
(b) the time taken to travel directly from O t o B .
(AEB)
6. A particle performs simple harmonic motion
about a point O on a straight line. The period of
motion is 8 s and the maximum distance of the
particle from O i s 1.2 m. Find its maximum
speed and also its speed when it is 0.6 m from O.
Given that the particle is 0.6 m from O after one
second of its motion and moving away from O,
find how far it has travelled during this one
second.
(AEB) 204 7. A particle describes simple harmonic motion
about a centre O. When at a distance of 5 cm
from O its speed is 24 cm s −1 a nd when at a
distance of 12 cm from O its speed is 10 cm s −1 .
Find the period of the motion and the amplitude
of the oscillation.
Determine the time in seconds, to two decimal
places, for the particle to travel a distance of
3 cm from O.
(AEB)
8. A particle moves along the x axis and describes
simple harmonic motion of period 16 s about the
origin O as centre. At time t = 4 s, x = 12 cm
and the particle is moving towards O with speed
5π
cm s −1 . G iven that the displacement, x , at
8
any time, t , m ay be written as x = a cos( ω t + φ ),
find a, ω and φ . ( AEB) 9. A particle, P , o f mass m i s attached, at the point
C, to two light elastic strings AC and BC. The
other ends of the strings are attached to two
fixed points, A and B, on a smooth horizontal
table, where AB = 4 a. B oth of the strings have
the same natural length, a , a nd the same
modulus. When the particle is in its equilibrium
position the tension in each string is m g . Show
that when the particle performs oscillations
along the line AB in which neither string
slackens, the motion is simple harmonic with 2a period π . g
The breaking tension of each string has
3 mg
magnitude
. S how that when the particle is
2
performing complete simple harmonic
oscillations the amplitude of the motion must be
1
less than a.
2
Given that the amplitude of the simple harmonic
1
oscillations is a, f ind the maximum speed of
4
the particle.
(AEB) Chapter 8 Simple Harmonic Motion 10. The three points O , B, C lie in that order, on a
straight line l o n a smooth horizontal plane with
OB = 0.3 m, OC = 0.4 m.
A particle, P, describes simple harmonic motion
with centre O along the line l . At B the speed of
the particle is 12 ms −1 a nd at C its speed is
9 ms −1 . F ind 11. A particle, P , d escribes simple harmonic motion
in the horizontal line ACB, where C is the midpoint of AB. P is at instantaneous rest at the
points A and B and has a speed of 5 ms −1 w hen it
passes through C. Given that in one second P
completes three oscillations from A and back to
A, find the distance AB. Also find the distance
of P from C when the magnitude of the (a) the amplitude of the motion; acceleration of P is 9 π 2 ms −2 . (b) the period of the motion; Show that the speed of P when it passes through
the point D, which is the midpoint of AC, is (c) the maximum speed of P;
(d) the time to travel from O to C.
This simple harmonic motion is caused by a light
elastic spring attached to P. The other end of the
spring is fixed at a point A on l w here A is on the
opposite side of O to B and C, and AO = 2 m.
Given that P has mass 0.2 kg, find the modulus of
the spring and the energy stored in it when
AP = 2.4 m.
( AEB) 53
ms −1 . A lso find the time taken for P to
2
travel directly from D to A.
(AEB) 12. A particle moves with simple harmonic motion
along a straight line. At a certain instant it is
9 m away from the centre, O, of its motion and
has a speed of 6 ms −1 a nd an acceleration of
9
ms −2 .
4
Find
(a) the period of the motion;
(b) the amplitude of the motion;
(c) the greatest speed of the particle.
Given that at time t = 0, t he particle is 7.5 m
from O and is moving towards O, find its
displacement from O at any subsequent time, t ,
and also find the time when it first passes through
O.
(AEB) 205 Chapter 8 Simple Harmonic Motion 206 ...
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 Simple Harmonic Motion

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