1. Simple Harmonic Motion
Simple Harmonic Motion- By Aditya
Abeysinghe
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2. Consider an object moving round a circle with center O and
rotating with a uniform angular speed ω.
Acceleration at A is ω2r, and this acceleration is directed
along the radius AO.
Simple Harmonic Motion- By Aditya
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3. Hence the acceleration of P towards O is w2rCosAOC,
Which is also equal to w2rSinθ
But r Sinθ = PO = y say.
Therefore, acceleration of P towards O is w2y
Since w2 is a constant,
Acceleration of P towards O α distance of P from O.
Since the acceleration of the object is always
directed towards the center, acceleration should
be expressed as,
Acceleration towards O = - ω2y.
The minus sign indicates that the object begins to slow
down its motion as it passes the center of its path.
Simple Harmonic Motion- By Aditya
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4. Thus, Simple Harmonic Motion can be expressed
as follows:
It is the motion of any object
whose acceleration is always
directly proportional to the
distance from its center of path
and the acceleration is always
directed towards the center.
Simple Harmonic Motion- By Aditya
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5. The relationship between Simple Harmonic
Motion and Circular motion
Consider an object A rotating round its center of
path O.
amplitude
B
C
Period (one cycle)
Simple Harmonic Motion- By Aditya
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6. The B and C points show the corresponding
points where the circular motion is equivalent to
the wave. Since the circular motion occurs
continuously, it can be concluded that the
corresponding wave pattern also equals
periodically.
Thus Simple Harmonic Motion is the same
concept as moving an object in circular,
continuous motion.
Simple Harmonic Motion- By Aditya
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7. Amplitude of Motion
Consider an object attached to a spring as
shown below.
amplitude
X = Xmin
amplitude
X=0
X = Xmax
Direction of motion
Simple Harmonic Motion- By Aditya
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8. When you release the mass, the spring will
exert a force, pushing the mass back until the
object reaches the position x=xmax . This
position where the object has reached its
maximum is called its amplitude.
Theoretically,
Xmin = - Xmax .
Simple Harmonic Motion- By Aditya
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9. Hooke’s Law
Law: The force exerted on a body by an external
source is directly proportional to the mean
displacement the object has displaced from its
mean position.
Consider a spring-mass system as shown below.
F
X=0
X – displacement from mean position
Simple Harmonic Motion- By Aditya
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10. According to Hooke’s law,
F α x . (F-Force, X- Mean displacement)
However, theoretically the force on the object is always
acting so that the direction of force always opposes the
direction of displacement.
For example, when the spring travels to its amplitude
from its mean position, x increases while F decreases.
Similarly when the object returns to its original position
the force exerted on the object gradually increases while
the displacement from mean position decreases.
Thus, theoretically,
F α – X.
Simple Harmonic Motion- By Aditya
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11. To eliminate the proportionate sign, we use a constant k,
which is called the spring constant (since it depends on
the spring used).
Thus, F= -kX.
Since, F=ma, by Newton’s 2nd law of motion, the
direction of F is the direction of acceleration.
Thus, like acceleration, the force that acts on an
object in simple harmonic motion is always
directed towards the center of its path.
Since, this force acts towards equilibrium,
it is called a restoring force.
Simple Harmonic Motion- By Aditya
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12. Displacement of an object in Simple Harmonic
Motion
t=0
Sin θ = x/A
X = A Sinθ
θ = ωt
ω
X
t=t
θ
A
X = A Sin ωt
However, if the object when released
(at time=0) is not at the center of its path,
then the displacement of the object
is not accurate to be displaced as above.
Thus for accuracy, the displacement equation
is generally expressed as,
X
X = A Sin (ωt + φ), where φ is the phase lag.
Simple Harmonic Motion- By Aditya
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13. Period in Simple Harmonic Motion
Since the simple harmonic motion is analogous
to circular motion the period of an object in
simple harmonic motion is the same as that in
circular motion.
Thus, the period is the,
The time taken by an object to complete one
total rotation.
Thus, this can be graphically expressed as,
Simple Harmonic Motion- By Aditya
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14. Since θ for one complete revolution is 2π,
And that ω = θ/t ,
ω = 2π/t or the period of motion,
T = 2π / ω.
However, f = 1/ T, where f is the frequency of motion, or the
no. of rotations per unit time,
It is evident that, f = ω / 2π or ω
= 2πf.
Simple Harmonic Motion- By Aditya
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15. Relationship between velocity and displacement
V = V’ Cosθ
But,
Cos θ = √(A2 – x2)
θ
V’
A
And, V’= ωr
Since r = A or the amplitude,
V’ = ωA
Thus, from the above derivations,
V= ωA Cosθ or V= ωA √(A2 – x2)
Thus,
V= ω √(A2 – x2)
ω
X
θ
A
V
√(A2- X2)
X
Simple Harmonic Motion- By Aditya
Abeysinghe
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16. Potential energy of a spring
The potential energy of a spring, also sometimes called the elastic
potential energy can be calculated as follows:
U=-∫
F dx.
However, we know that F = -kX.
∫ (-kx) dx = ∫ kx dx
= k ∫x dx = k { x2/2}
Thus, U = -
Thus, U=
½
2
kx
Simple Harmonic Motion- By Aditya
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17. Oscillating Systems- Spring and
Mass
From Hooke’s law,
F = -kX
From Newton’s second
law of motion,
F = ma
F
X
Simple Harmonic Motion- By Aditya
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18. Therefore, ma = - kx
Thus, a = - (k/m) x.
However, we know that for any object
experiencing simple harmonic motion,
a= -ω2x. Therefore, -ω2x = - (k/m) x.
Thus, ω2 = k/m.
Therefore, the period of this motion, T,
T = 2π/ω = 2π/ {√(k/m)}
T = 2π √m/k
Simple Harmonic Motion- By Aditya
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19. Principle of conservation of
energy in simple harmonic motion
By applying principle of conservation of energy,
Kinetic energy
= Potential energy
at any position
at any position
Thus, ½ mr2ω = ½ kx2
Simple Harmonic Motion- By Aditya
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20. Oscillation of springs
Oscillations of springs can be described under
two categories based on the mode of oscillation
as,
1. Horizontal oscillations
2. Vertical oscillations
The main difference between the two types of
oscillations is that in vertical oscillations the
gravitational potential energy too contributes
towards(in addition to elastic potential energy and
kinetic energy, as in horizontal oscillations) the total
energy at any moment in the motion.
Simple Harmonic Motion- By Aditya
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21. Horizontal oscillation of springs
At xmin and at xmax the object gains
the maximum potential energy, as
the object has reached its
amplitude of motion.
Xmin
X=0
Xmax
The object gains the maximum
velocity, hence the maximum
kinetic energy as it passes the
mean position or the center of its
path
The total energy of the
spring-mass composite
= elastic potential
energy + kinetic energy.
The total energy is conserved
for the system or the system
obeys conservation of
mechanical energy
Simple Harmonic Motion- By Aditya Abeysinghe
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22. Vertical oscillation of springs
Similar to that in horizontal oscillation , the
object gains maximum elastic potential energy
at its amplitude and maximum kinetic energy
at its center of path.
However, total energy of the
spring-mass composite = elastic
potential energy + gravitational
potential energy + kinetic energy
Simple Harmonic Motion- By Aditya Abeysinghe
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23. Equilibrium Position
Because the mass exerts a gravitational force to
stretch the spring downward, the equilibrium
position is not the position where x = 0, but,
x = -h, where h is the vertical displacement of
the spring due to gravity.
The equilibrium position is the place
where the net force acting on the
object is zero
Simple Harmonic Motion- By Aditya
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24. Initial Equilibrium Position
(Position without weight)
Final Equilibrium Position
(Position gained due to weight mg)
Restoring force acting on the spring = -kh
The gravitational force acting on the object =
mg
However, for the object to be stationary, by applying F = ma for the
composite a = o and thus the restoring force = gravitational force.
Thus, -kh = mg or h = -(g/k)m.
Thus, it should be noted that hα m. (higher masses drag the spring
further towards gravity)
Simple Harmonic Motion- By Aditya Abeysinghe
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25. Coupled Spring Systems
1. Springs in series
Consider two springs with spring constants k1 and k2
connected to a mass m. When m is displaced by a
distance x, and if the displacements made by the two
springs are x1 and x2 respectively,
Then x1 + x2 = x.
Simple Harmonic Motion- By Aditya
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26. If the restoring force is F, then
F = -k 1x1 = -k2 x2. Thus, x1 = -F/k
1
and x2= -F/ k2
Since, x1 + x2 = x , it can be derived that
X = {-F/k 1 } + {-F/ k2 }.
By applying Hooke’s law to the spring composite, we can
write it as,
F = -keff X , and hence X = -F/keff. (keff – keffective)
By substituting this to the above derivation, we
find that,
keff = k 1 k2 /(k 1 + k2)
And period of oscillation,T,
T = 2π √ m { (k 1 + k2)/ k 1 k2 }
Simple Harmonic Motion- By Aditya
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27. 2. Springs in parallel
In this case if m is displaced by a distance of
x, both springs display by x. However, the force
exerted by the two springs on m on extension
are unequal.
For first spring,
F1 = -k1x
For second spring,
F2 = -k2 x
Simple Harmonic Motion- By Aditya
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28. Since the total force on m is equal to the
summation of the individual forces exerted by
the two springs,
F = F1+ F2
Thus,
-keff x = (-k1x )+ (-k2 x)
Therefore,
keff = k1 + k2
The period of motion, T,
T = 2π √ m / (k1 + k2)
Simple Harmonic Motion- By Aditya
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29. 3. Masses connected between two strings
If the body is displaced, one of the strings get extended
while the other gets compressed.
Thus, the total force on the object is the sum of the two
restoring forces exerted by the two springs.
Thus, F = F1+ F2
As in the case of springs in parallel, keff = k1 + k2
Simple Harmonic Motion- By Aditya
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30. Pendulums
θ
F’
mg Cosθ
mg Sinθ
F = ma = mg Sinθ
However, for small angles, Sinθ = θ, and, θ = s/l
Thus, F = mgs/l
Therefore, ma = mgs/l.
Thus, a = gs/l.
Since, the force is in the opposite direction of the displacement. It should be written
as ,
a = -gs/l or a = -(g/l) s
This is in the form a = -ω2x and the pendulum hence shows a Simple Harmonic
Motion, where, ω
= g/l.
Simple Harmonic Motion- By Aditya
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31.  Period of motionT = 2π/ω . Therefore, T
= 2π √(l/g)
 EnergyThe mechanical energy of a pendulum is a conserved quantity.
θ
θ = θmin
θ=0
θ = θmax
Velocity = 0
Velocity= max
Velocity = 0
Potential energy = max
Potential energy = min
Potential energy = max
Restoring force = max
Restoring force = 0
Restoring force = max
Tension = min
Tension = max
Tension = min
Kinetic energy = 0
Kinetic energy = max
Kinetic energy = 0
Simple Harmonic Motion- By Aditya Abeysinghe
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32.  VelocityX
L
θ
h
V
X = L Cosθ
Therefore, h = L - L Cosθ
From the conservation of energy,
Total initial energy = Total final energy
½ mv2 = mgL (1- Cos θ )
Thus, V
V αL
Vαθ
= √{2gL (1- Cos θ )}
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