1

SIMPLE PENDULUM
Overview
A body is said to be in a position of stable equilibrium if, after displacement in
any direction, a force restores it to that position. If a body is displaced from
a position of stable equilibrium and then released, it will oscillate about the
equilibrium position. We shall show that a restoring force that is proportional
to the displacement from the equilibrium position will lead to a sinusoidal os-
cillation; in that case the system is called a harmonic oscillator and its motion
is called a harmonic motion. The simple harmonic oscillator is of great impor-
tance in physics because many more complicated systems can be treated to a
good approximation as harmonic oscillators. Examples are the pendulum, a
stretched string (indeed all musical instruments), the molecules in a solid, and
the atoms in a molecule.[2]
Denition
A hypothetical apparatus consisting of a point mass suspended from a weight-
less, frictionless thread whose length is constant, the motion of the body about
the string being periodic and, if the angle of deviation from the original equi-
librium position is small, representing simple harmonic motion.[3]
OR
A simple pendulum consists of a bob suspended by a cord. The bob, in
general, is spherical and has a nite mass. The cord, though exible, is un-
stretchable; it does not get stretched by the weight of the bob or by its swinging
motion.[4]
Theory
A simple pendulum consists of a mass m suspended from a xed point by a
string of length L. If the mass is pulled aside and released, it will move along
the arc of a circle, as shown in g. Let s be the distance from the equilib-
rium position measured along that arc. While the force of gravity, mg, points
downward, only its tangential component along the arc,
F
tan
= mgsin()
2
acts to accelerate the mass. The minus sign indicates that F
tan
is a restoring
force, i.e. one that points in a direction opposite to that of the displacement
s. From Newtons second law,
F
tan
= ma
we then get,
mgsin() = ma (1)
By Denition
a =
d
2
s
dt
2
So (1) implies
d
2
s
dt
2
= gsin()
We can easily express s in terms of by noting that
s = L
Hence
d
2
s
dt
2
= L
d
2

d
2
and thus,
d
2

d
2
=
gsin
L
(2)
This second order dierential equation is the equation of motion for the pen-
dulum. It looks deceptively simple but is actually quite dicult to solve. For
many applications, including ours, a simplied approximation is sucient. We
can obtain the approximate solution by using the series expansion
sin =

3
3!
+

5
5!

From this series expansion, we see that we can approximate sin by the angle
itself, as long as we keep so small that the higher order there are much
smaller than .
By using
sin ~
Equation (2) becomes,
d
2

d
2
=
g
L
(3)
3
This is the equation of motion of a simple harmonic oscillator, describing a
simple harmonic motion. It is solved for arbitrary values of the amplitude,

max
and the phase, by
=
max
cos(t + ) (4)
If you substitute Eq. (4) into Eq. (3), you will nd that the angular frequency,
, is not arbitrary. It must have the value
=

g
L
(5)
The cosine oscillates between 1 and 1; hence, according to Eq. (4), the angle
will oscillate over time between
max
and
max
. The time to execute one
complete cycle is called the period T and is related to by
T =
2

(6)
The speed of the mass is given by
v =
ds
dt
= v = L
d
dt
s = L
If we substitute from Eq. (4) we obtain
v(t) =
max
Lsin(t + )
=
max
Lsin(t + +

2
)
(7)
Comparing Eq. (7) with Eq. (4), we see that the speed v oscillates with the
same frequency as the angle but is out of phase with by

2
. Clearly this
makes sense; the speed is zero when the angle is greatest.[2]
Energy analysis of the pendulum :
For a pendulum swinging back and forth, the mechanical energy, E, shifts
between kinetic and potential energy, but remains constant:
E = K + U
U = mgL
K =
1
2
mv
2
Here L is vertical displacement from equilibrium, and v is velocity of the bob.
When the bob is at the maximum amplitude, x = x
m
, and L = h (the maximum
4
vertical displacement). At this point, v = 0: there is no kinetic energy, so all
the energy is potential energy. The bob has greatest speed at its lowest point,
hence all the energy is kinetic, and U = 0.
Conservation of mechanical energy for these two instants can be expressed as:
K
0
+ U
0
= K
m
+ U
m
where subscript 0 stands for values evaluated at the equilibrium position (x =
0, y = 0) and the subscript m stands for values at highest point of the oscillation
(x = xmax, L = h) . Then we can evaluate each term and nd
1
2
mv
2
0
+ 0 = 0 + mgh (8)
This equation relates the maximum velocity (at x=0) to the maximum height
and the value of g. It also allows us to express the total kinetic energy to the
height under the assumption of energy conservation. Curiously, the maximum
velocity is achieved at the equilibrium position! We can also solve (8) for v
2
0
and obtain:
v
2
0
= (
x
t
)
2
= 2gh
This expression will be useful when we study energy conservation.
The period of pendulum without small angle approximation:
Let 0 in g. be the xed point of support, OP the length of pendulum , and
m the mass of particle at P.
In polar coordinates the position of particle
r = r r
Dierentiating
r = v = r r + r

r
v = r r + r

(9)
and

v = a = r r + r

r + r

+ r

+ r

a = ( r r

2
) r + (r

+ 2 r

)

(10)
This gives, for the force along radial vector F
r
F
r
= m( r r

2
)
5
and this is equal to the components of applied force acting on mass m in the
direction OP. Thus
m( r r

2
) = mgcos T
where T is the tention in the cord at the angle . Similarly, for the transverse
force F

in the direction of increasing , we have from Eq. (10)

= (r

+ 2 r

) = mgsin
Since the string is assumed to be inextensible, the vector r has a constant
magnitude equal to l, and it follows that r = 0 and r = 0
The equation of motion becomes
ml

2
= mgcos T (11)
ml

mgsin (12)
To make Eq.(12) linear, let us assume that is so small that sin may be set
equal to in radian measure. This really assumed that
3
and higher powers
of are negligibly small compared to , since the expression of sin in radian
measure is
sin =

3
3!
+

5
5!

If therefore the maximum angular amplitude is a few degree, we may write
Eq.(12) as

=
g
l

This is the general form of a dierential equation for simple harmonic motion
whose period is:
P = 2

l
g

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SOLVED EXAMPLES
QUESTION A lamp is suspended from a high ceiling with a cord 12ft long.
Find its period of oscillation.
SOLUTION
12ft = 3.6576m
We know that time period of pendulum is given by:
T = 2

L
g
= 2

3.6576m
9.98m]s
2
= 3.85s
QUESTION The period of a simple pendulum is directly proportional to the
square root of its length, If a pendulum has a length of 6 feet and a period of
2 seconds, to what length should it be shortened to achieve a 1 second period?
SOLUTION The period of a simple pendulum is directly proportional to the
square root of its length.
Period = T
Length = L
T = K

L
where k is a constant of proportionality. to work out what k is, rearrange the
equation above:
K =
T

L
originally,
T = 2
L = 6
this gives
K =
2

6
So now we can write
T = (
2

6
).

L
7
we want T to be 1, so:
1 = (
2

6
).

6
2
=

L
= L =
6
4
ft
QUESTION Using a small pendulum of length 0.171m, a man counts 72.0
complete swings in a time of 60.0 s. What is value of g in this location?
SOLUTION
T =
time
no. of oscillations
=
60s
72
= 0.833s
We know that
T = 2

L
g
= T
2
= 4
2
L
g
= g =
4
2
L
T
2
=
39.5 0.171m
(.833s)
2
= g = 9.73m]s
2
QUESTION A simple pendulum of length 2m oscillate too and fro. What is
time period of this pendulum?
SOLUTION
We know that time period of simple pendulum is given by formula
T = 2

l
g
T = 2 3.14

2m
9.98m]s
2
T = 2.81s
QUESTION Let a simple pendulum having time period 3.24s to complete
its oscillation. Find the length of pendulum.[1]
SOLUTION
T = 2

l
g
8
Squaring on both sides
T
2
= 4
2
l
g
= l =
gT
2
4
2
Substituting values in above equation
l =
9.98 (3.24)
2
4 (3.14)
2
= l =
104.76
39.43
= 2.6568m
Hence required length of pendulum is 2.6568m

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References
[1] Dr.M.Fogiel,A The Mechanics Problem Solver ,Trans .Amer .Math
soc.226(1977) ,(257-290).
[2] S.M.Trimzi,Mechanics, Karvan Book House.
[3] http://dictionary.reference.com/browse/simple+pendulum
[4] http://www.pa.msu.edu/