Subject: Physics
Lesson 3: Mechanical Oscillator
Class: Grade 12-LS
Section: A & B
Done By Mr. Nawaf AL Souheil
Objectives
-Quick review for the trigonometric functions
-Quick review for the differential equation
-Define the horizontal oscillator
-Energetic study for the Simple Harmonic motion (SHM)
-Define the periodic motion
-Dynamic study for the simple harmonic motion (SHM)
-Solution for the differential equation
Trigonometric Function
Sin(x) & Cos(x)
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4
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.�These are the maximum points, the minimum points, and the intercepts.
0
-1
0
1
0
sin x
0
x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y
x
y = sin x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.�These are the maximum points, the minimum points, and the intercepts.
1
0
-1
0
1
cos x
0
x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y
x
y = cos x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y
x
Example: Sketch the graph of y = 3 cos x on the interval [–π, 4π].
Partition the interval [0, 2π] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min
x-int
max
3
0
-3
0
3
y = 3 cos x
2π
π
0
x
(0, 3)
( , 0)
( , 0)
( , )
( , – )
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7
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
x
y = – 4 sin x
reflection of y = 4 sin x
Is y= - 4sinx
y = 4 sin x
y = 2sin x
y = sin x
y = sin x
Differential equation
A differential equation is any equation with a variable and its either first derivative or second derivative, or higher order derivative.
Mathematical study of a second order differential equation:
In physics all quantities of motion are studied with respect to time, and the position is denoted by x, thus the differential equation that we will study will have the form:
Horizontal Oscillator
A horizontal oscillator is formed of a horizontal spring of stiffness K (in N/m), fixed at one end, while the other end a mass m is attached to it. (see the figure)
The motion of this oscillator is called: Simple Harmonic motion (SHM)
Simple Harmonic Motion (SHM):
Periodic motion:
Is the motion that repeats itself within a specific time interval called the period T of the motion.
Example: the second’s arm in the clock repeats its cycle each 1 minute, thus its period is T=1 min=60 s
Example: the harmonic motion repeats itself within its period that is defined by:
(This relation will be derived later)
Energetic study of Horizontal Oscillator
Consider a horizontal oscillator formed of a spring with force constant K, and a block of mass m, placed as shown in the figure (Assume that the friction force is neglected):
At t=0, the system is released, and thus it begins to oscillate as shown in the figure.
The Motion of the system is called: Simple Harmonic motion (SHM)
At t=0, the system is storing a pure elastic energy when the reference for the gravitational potential energy is taken along the horizontal axis of the oscillator.
Thus at t=0, the mechanical energy of the system (Spring,block,Earth) is given by:
At any instant t, the system is oscillating, and it possesses both kinetic and elastic potential energies.
Thus at any instant t, the mechanical energy of the system is given by:
Since the friction force is neglected, so the mechanical energy is conserved, then:
Derive the obtained equation with respect to time t:
Relation between oscillations and sinusoidal functions
Dynamic Study:
Forces acting on the system during its motion in the absence of friction force are:
Applying Newton’s Second law: