Physics
Lecture Summary
1. Oscillatory motion
1.1 Simple harmonic oscillation oscillation One of the most frequent motions in nature is oscillatory or vibrational motion. A particle is oscillating when it moves periodically about an equilibrium position. The oscillatory motion is the result of the so called linear restoring force. This is a force whose magnitude is proportional to the displacement of a particle from some equilibrium position, and the direction is opposite to that of the displacement. Such a force is exerted by an elastic cord or by a spring obeying Hooke’s law: F x = − Dx, D > 0 The proportionality factor D is called stiffness, or force constant of the spring. Suppose that the straight line of the motion is the x-axis. Apply Newton’s II law. The equation of motion: ii
m x = − Dx Solve this second order differential equation for the x(t) function: D x = − x , m Introduce the next symbol: D ω 02 = m The differential equation: ii
ii
x = −ω 02 x , ii
x + ω 02 x = 0
It is a second order homogeneous linear differential equation with constant coefficients. The general solution is the linear combination of the two independent partial solutions like: x1 = sin ω 0t , x2 = cos ω 0t .
The linear combination: x(t ) = C1 sin ω0 t + C2 cos ω 0t , or x (t ) = A sin (ω 0t + δ ) .
In the first general solution the two integration constants are C 1 , and C 2 . In the second solution the two integration constants are A and δ . The displacement x is a sinusoidal function of the time t . The coefficient ω 0 is called angular frequency. The maximum value of the displacement is called amplitude of the oscillation A. As the function repeats itself after a 2π time so this is the period of the motion T . ω 0 T =
2π ω 0
= 2π
m D
The time of period is determined completely by the mass of the oscillating particle and the stiffness of the spring. The frequency of the oscillation is the number of cycles in unit time: B. Palásthy
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Lecture Summary
f 0 =
1
T The quantity (ω 0t + δ ) is called phase of the oscillation and this δ is the initial phase. The
amplitude A and the initial phase can be obtained from the initial condition. The plot of the displacement versus time function is shown on the figure: x A T t
− A displacement time function: x = A sin (ω 0t + δ )
velocity time function: i
x = Aω 0 cos (ω 0t + δ )
acceleration time function: ii
x = − Aω 02 sin (ω 0t + δ ) .
The linear restoring force is a conservative force. To obtain the potential energy we apply:
F = −∇ V
In one dimension:
=− F x
∂V , but the linear restoring force F x = − Dx ∂ x ∂V − Dx = − ∂ x V =
1
Dx 2 + C
2 Choosing the zero of the potential energy at the equilibrium position, we get: C = 0 1 2 V ( x ) = Dx 2 In case of conservative force the mechanical energy remains constant: E = T + V = constant 2 1 1 1 1 2 2 2 2 2 2 m x + Dx = mA ω 0 cos (ω 0t + δ ) + DA sin (ω 0t + δ ) = 2 2 2 2 1 1 = DA2 cos 2 (ω 0t + δ ) + DA2 sin 2 (ω 0t + δ ) = 2 2 1 1 = DA2 ⎡⎣sin 2 (ω 0t + δ ) + cos 2 (ω 0t + δ ) ⎤⎦ = DA2 2 2 During an oscillation, there is a continuous exchange of kinetic and potential energies. i
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Lecture Summary
1.2 Superposition of two simple harmonic motion
1.2.1 Same direction, same frequency Consider the superposition, or interference of two simple harmonic motions, that produce a displacement of the particle along the same line. Suppose that both have the same frequency. The displacement time functions are given by: x1 = A1 cos (ωt − ϕ 1 ) x2 = A2 cos (ωt −ϕ 2 )
The resulting displacement is given by: x = x1 + x2 = A1 cos (ωt − ϕ1 ) + A2 cos (ωt −ϕ 2 ) = = A1 cos ωt cos ϕ1 + A1 sin ωt sin ϕ1 + A2 cos ωt cos ϕ 2 + A2 sin ωt sin ϕ 2
x (t ) = cos ωt [ A1 cos ϕ1 + A2 cos ϕ 2 ] + sin ωt [ A1 sin ϕ1 + A2 sin ϕ 2 ]
So the result is a harmonic oscillation with the same frequency: x (t ) = A cos (ωt − ϕ ) = A cos ωt cos ϕ + A sin ωt sin ϕ
where A cos ϕ = A1 cos ϕ1 + A2 cos ϕ 2 ⎫ ⎪⎪
⎬
A sin ϕ = A1 sin ϕ1 + A2 sin ϕ 2 ⎪ ⎪⎭
solving this system for A and ϕ : tan ϕ =
A =
A1 sin ϕ1 + A2 sin ϕ 2 A1 cos ϕ1 + A2 cos ϕ 2
, and
A12 + A22 + 2 A1 A2 cos (ϕ 2 −ϕ 1 )
The amplitude of the resultant oscillation depends on the amplitudes of the two original amplitudes and on the difference of the two original initial phases. Let us consider some important special cases: 0 The two motions are in phase. The amplitude of the interference is: If ϕ1 = ϕ 2 ⇒ ϕ 2 −ϕ1 = A =
A12 + A22 + 2 A1 A2 cos 0 = A1 + A2 1
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Lecture Summary
Hence the two simple harmonic motions interfere by reinforcement because their amplitude adds. If ϕ1 −ϕ 2 = π ⇒ , the two oscillations are in opposite phase, and the amplitude: A =
A12 + A22 + 2 A1 A2 cos π = A1 − A2 −1
In this case we say, that the two simple harmonic motions are in opposition, and the two simple harmonic motions interfere by attenuation, because their amplitudes subtract. If the two amplitudes are equal A1 = A2 , they completely cancel each other.
1.2.2 Same direction, different frequency We shall now consider the case of two interfering simple harmonic oscillations, in the same direction but different frequencies. We assume that the initial phase of the first oscillation is zero. x1 = A1 cos ω 1t x2 = A2 cos (ω2t − ϕ )
In this case in general the resultant motion will not be periodical. An interesting situation occurs when A1 = A2 that is, when the two amplitudes are equal. x1 = A1 cos ω 1t x2 = A cos(ω2t − ϕ ) x = A ⎡⎣ cos ω1t + cos (ω2t − ϕ )⎤⎦
Some mathematical considerations: cos (α + β ) = cos α cos β − sin α sin β , cos (α − β ) = cos α cos β + sin α sin β The sum of the two equations: cos (α + β ) + cos (α − β ) = 2cos α cos β , if α + β = ω 1t , and α − β = ω2 t −ϕ then 2α = (ω1 + ω2 ) t − ϕ α =
ω1 + ω 2
t −
ϕ
, 2 2 2 β = ω1t − ω2t + ϕ
β =
ω1 − ω 2 2
t +
ϕ 2
.
So:
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Lecture Summary
⎛ ω1 − ω2 ⎛ ω + ω2 ϕ ⎞ ϕ⎞ t + ⎟⎟ cos ⎜⎜ 1 t − ⎟⎟ ⎟ ⎜ ⎝⎜ 2 ⎠ ⎝ 2 2 2 ⎠⎟
x = A ⎡⎣ cos ω1t + cos (ω2t −ϕ )⎤⎦ = 2 A cos ⎜⎜
If ω1 − ω2
min (ω1 , ω2 ) , it means that the two frequencies are different but close, the
resultant vibration is not harmonic. x
2A
0
t
-2 A
⎛ ω1 − ω2 ϕ ⎞ t + ⎟⎟ describes a very slow variation and we can include it into the ⎝ 2 2 ⎟⎠
The term cos ⎜⎜ ⎜
amplitude. This time varying amplitude is shown by the dashed line. The situation described arises when, for example, two tuning forks of close but different frequencies are vibrating simultaneously at nearby places. The fluctuation in the intensity of the sound is called beats. The fluctuation is due to the change in amplitude. The amplitude is “modulated”. If the two amplitudes are not equal x1 = A1 cos ω 1t , x2 = A2 cos (ω2t −ϕ ) .
The superposition of the two oscillations is shown on the next figure: x A 1 + A 2
a b s ( A 1 -A 2 ) 0
t
The calculation is more difficult but the result is similar as in previous case. The amplitude is modulated between: A1 + A2 and A1 − A2 .
1.2.3 Fourier Analysis of Periodic Motion Consider now the superposition of the next two harmonic oscillations:
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Lecture Summary x1 = A sin ω t , x2 = B sin2ω t .
The two angular or cyclic frequencies are ω and 2ω . The superposition is periodic but not simple harmonic. x
A
B
0
t
T
x = A sin ωt + B sin 2ω t
If we add terms of C sin 3ωt , D sin 4ω t..... and so on we still get a displacement x that is periodic. The frequency ω is called fundamental, and the frequencies 2ω, 3ω, 4ω ... are the harmonics or overtones. The time of period of x is the same as the time of period of the fundamental oscillation: ω =
2π T
→ T =
2π ω
By adding simple harmonic motions whose frequencies are multiples of a fundamental frequency and whose amplitudes are properly selected, we may obtain almost any arbitrary periodic function. The reverse is also known and constitutes Fourier’s theorem proved in mathematics. Fourier’s theorem asserts that a periodic function f (t ) can be expressed as a sum: f (t ) = A0 + A1 cos ωt + A2 cos 2ωt + ... An cos nωt + ... + B1 sin ωt + B 2 sin 2ωt + ... Bn sin nωt + ...
This formula is known as Fourier series. One more cause why single harmonic motion is important.
1.2.4 Perpendicular directions, same frequency Consider a particle moving in a plane in such a way that the particle’s two coordinaters x and y oscillate with simple harmonic motion and have the same frequency. x = A cos ω t , y = B cos (ωt − ϕ )
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Lecture Summary
This equation system is called the parametric equation system of the trajectory. ϕ is the phase difference of the two oscillations. We can obtain the equation of the trajectory if we eliminate the time. x A y
/⋅ cos ϕ
= cos ωt
= cos ωt cos ϕ + sin ω t sin ϕ .
B
Subtract the two equations: x A
y
cos ϕ −
B
= − sin ωt sin ϕ
The square: x
2 2
2
A
cos ϕ +
y
2
B
2
−
2 xy
cos ϕ = sin 2 ω t sin 2 ϕ
AB 2 2 x y xy 2 cos ϕ + 2 cos ϕ = (1− cos 2 ω t )sin 2 ϕ − 2 2 A B AB
⎛ x 2 ⎞⎟ 2 ⎜ cos ϕ + 2 − 2 cos ϕ = ⎜1− 2 ⎟sin ϕ ⎜⎝ A2 B AB A ⎟⎠ x 2
x
y2
2
2 2
2
A
cos ϕ +
x
xy
2
A
2
sin ϕ + 2
y B
2
− 2
2 xy AB
2 cos ϕ = sin ϕ ,
Finally we obtain: x
2 2
A
+
y B
2
− 2
2 xy AB
2 cos ϕ = sin ϕ
This is just the equation of a general ellipse. Consider now the special cases a) if , ϕ = 0 there is no phase difference between the two oscillations:
⎛ x y ⎞2 + 2− = 0 → ⎜⎜ − ⎟⎟ = 0 2 ⎝⎜ A B ⎠⎟ A B AB x 2
y2
2 xy
y =
B A
x,
this is the equation of a straight line, whose slope is positive
B A
.
b) if ϕ = π the phase difference: 2
⎛ x y⎞ ⎜⎜ + ⎟⎟ = 0 → + + = 0 ⎜⎝ A B ⎟⎠ A2 B 2 AB x 2
y2
2 xy
B y = − x, A
also a straight line with negative slope. Therefore we say that when ϕ = 0 or π , the interference of two perpendicular simple harmonic motions of the same frequency results in rectilinear polarization.
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c) if ϕ =
Lecture Summary π 2
, the phase difference is
π 2
between the two oscillations: x 2
y2
+ = 1, A2 B 2 this is the equation of a special ellipse, the axes are parallel with the x, and y coordinate axes. In this case the particle moves along the ellipse in clockwise direction. If A = B the amplitudes are equal the resulting motion is circular.
d) if ϕ =
3π 2
, the equation is also an ellipse: x 2
y2
+ =1, A2 B 2 but the motion is counter clockwise. Thus we may say that when then phase difference is π ± , the interference of two simple harmonic motions of the same frequency results in 2 elliptical polarization with the axes of the ellipse parallel to the directions of the two motions.
e) for an arbitrary value of the phase difference ϕ , the path is still an ellipse, but its axes are rotated relative to the coordinate axes. the different cases: y
y B
B
− A
O A
− A
x
− B y
− A
y
ϕ =
π 2
− A
A x
O
− B
x
ϕ = π − B
ϕ = 0
B
A
O
ϕ =
3π 2
A
ϕ =
O
A
− A
ϕ =
π 2 x
3π 2
y B
− A
O
A
x
− B ϕ = arbitrary
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Lecture Summary
1.2.5 Perpendicular directions, different frequencies x = A cos ω 1t , y = B cos (ω2t + ϕ )
In general case the superposition is not periodic motion. If the two component frequencies ω 1 and ω 2 are the ratio of two integers that is rational number, we can obtain the so called Lissajous figures. A simple way to produce such patterns is by means of an oscilloscope. The electron beam is deflected by each of two electric fields at r ight angles. The field strengths alternate sinusoidally. This part of physics is important on the fields: - polarized light - alternating current circuits - diffraction and interference of light, sound and electromagnetic radiation.
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