LECTURE ONE 2025 SEM II simple harmonic.ppt

Lecture Notes

1. 8/15/2025
PHY 212: OSCILLATIONS, WAVES AND OPTICS
LECTURE ONE: OSCILLATORY MOTION
THE UNIVERSITY OF GOROKA
SCHOOL OF SCIENCE AND TECHNOLOGY
DIVISION OF NATURAL AND PHYSICAL SCIENCES
COMPILED BY: ALEX W JOHN
1

2. OSCILLATORY MOTION
• Periodic motion
• Spring-mass system
• Differential equation of motion
• Simple Harmonic Motion (SHM)
• Energy of SHM
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OUTLINES:
COMPILED BY: ALEX W JOHN 2

3. PERIODIC MOTION
•Periodic motion is a motion that regularly returns to a given position after
a fixed time interval.
•A particular type of periodic motion is “simple harmonic motion,” which
arises when the force acting on an object is proportional to the position of
the object about some equilibrium position.
•The motion of an object connected to a spring is a good example.
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4. RECALL HOOKE’S LAW
• Hooke’s Law states F
s
= -kx
•F
s is the restoring force.
•It is always directed toward the equilibrium position.
•Therefore, it is always opposite the displacement from
equilibrium.
•k is the force (spring) constant.
•x is the displacement.
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5. 8/15/2025
Restoring Force and the Spring Mass System

 In a, the block is displaced to the right of x = 0.
The position is positive.
The restoring force is directed to
the left (negative).
 In b, the block is at the equilibrium position.
x = 0
The spring is neither stretched nor
compressed.
The force is 0.
In c, the block is displaced to the left of x = 0.
The position is negative.
The restoring force is directed to
the right (positive).

6. DIFFERENTIAL EQUATION OF MOTION
•Using F = ma for the spring, we have
•But recall that acceleration is the second derivative of the position:
•So this simple force equation is an example of a differential equation,
•An object moves in simple harmonic motion whenever its
acceleration is proportional to its position and has the opposite sign
to the displacement from equilibrium.
CONDITION!
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ma kx
2
2
d x
a
dt

2 2
2 2
or
d x d x k
m kx x
dt dt m
 

7. ACCELERATION
•Note that the acceleration is NOT constant, unlike our earlier
kinematic equations.
•If the block is released from some position x = A, then the initial
acceleration is – kA/m, but as it passes through 0 the acceleration
falls to zero.
•It only continues past its equilibrium point because it now has
momentum (and kinetic energy) that carries it on past x = 0.
•The block continues to x = – A, where its acceleration then
becomes +kA/m.
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8. ANALYSIS MODEL, SIMPLE HARMONIC MOTION
•What are the units of k/m, in ?
•They are 1/s
2
, which we can regard as a frequency-squared, so let’s
write it as
•Then the equation becomes
•A typical way to solve such a differential equation is to simply search for
a function that satisfies the requirement, in this case, that its second
derivative yields the negative of itself! The sine and cosine functions
meet these requirements.
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2
2
d x k
a x
dt m
 
2k
m

2
a x

9. SHM GRAPHICAL REPRESENTATION
•A solution to the differential
equation is
•A,  are all constants:

A = amplitude (maximum position
in either positive or negative x direction,

= angular frequency,
 = phase constant, or initial phase angle.
A and  are determined by initial conditions.
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() cos( )x t A t  
k
m Remember, the period
and frequency are:
2 1

2
T f
T
 
 
 
  
 
 
COMPILED BY: ALEX W JOHN 9

10. MOTION EQUATIONS FOR SHM
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2
2
2
( ) cos( )
( ) sin( )
( ) cos( )
x t A t
dx
v t A t
dt
d x
a t A t
dt
 
  
  
 
  
  
The velocity is 90
o
out of phase with the
displacement and the acceleration is
180
o
out of phase with the displacement.
COMPILED BY: ALEX W JOHN 10

11. CONSIDER THE ENERGY OF SHM OSCILLATOR
•The spring force is a conservative force, so in a frictionless system
the energy is constant
•Kinetic energy, as usual, is
•The spring potential energy, as usual, is
•Then the total energy is just
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21
2
(a constant)E K U kA  
 
2 2 21 1
2 2
cosU kx kA t    
 
2 2 2 21 1
2 2
sinK mv m A t    

12. TRANSFER OF ENERGY OF SHM
•The total energy is constant at all times, and is (proportional
to the square of the amplitude)
•Energy is continuously being transferred between potential energy
stored in the spring, and the kinetic energy of the block.
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21
2
E kA

13. PRACTICE PROBLEM
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14. Problem Two
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15. 8/15/2025 COMPILED BY: ALEX W JOHN 15
TUTORIAL QUESTION:

16. CASE STUDY!!
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Palanisamy, Gowri. (2017). A Case Study on Simple Harmonic Motion and
Its Application. International Journal of Latest Engineering and
Management Research (IJLEMR). Volume 02. 55-64.
Abstract
In this paper, we are going to study about simple harmonic motion
and its applications. The simple harmonic motion of a spring-mass
system generally exhibits a behavior strongly influenced by the
geometric parameters of the spring. In this paper, we study the
oscillatory behavior of a spring-mass system, considering the
influence of varying the average spring diameter Φ on the elastic
constant k, the angular frequency ω, the damping factor γ, and the
dynamics of the oscillations. Simple harmonic motion and obtains
expressions for the velocity, acceleration, amplitude, frequency and
the position of a particle executing this motion. Its applications are
clock, guitar, violin, bungee jumping, rubber bands, diving boards,
earthquakes, or discussed with problems.

17. 8/15/2025 COMPILED BY: ALEX W JOHN 17
End of Lecture One!