Simple Harmonic Motion In Physics, Composition Of Two SHM, Important Topics For JEE 2025
Shrivastav 26 May, 2024
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Simple Harmonic Motion In Physics : We will learn that A simple harmonic motion is produced when a restoring force proportional to the displacement acts on a particle. If the particle is acted upon by two separate forces each of which can produce a simple harmonic motion, the resultant motion of the particle is a combination of two simple harmonic motions.
Composition Of Two Shm Of The Same Period Along The Same Line
Composition Of Two Shm Of The Same Period Along The Same Line : The composition of two simple harmonic motions (SHMs) of the same period along the same line results in another SHM with the same period
Let the two SHM’s be
y 1 = A 1 sin ω t and y 2 = A 2 sin (ω t + ϕ)
The resultant displacement
y = y 1 + y 2 = A 1 sin ω t + A 2 sin (ω t + ϕ)
y
=
A
1
sin ω
t
+
A
2
sin ω
t
cos ϕ +
A
2
cos ω
t
sin ϕ
y
= (
A
1
+
A
2
cos ϕ) sin ω
t
+
A
2
sin ϕ cos ω
t
…(1)
Let A 1 + A 2 cos ϕ = R cos θ and A 2 sin ϕ = R sin θ.
Substituting in (1), we get
y = R [cos θ sin (ω t ) + sin θ cos (ω t )]
Thus, the resultant motion is also simple harmonic along the same line and has the same time period. Its amplitude R is
Composition Of Two Shm Of Same Period At Right Angles To Each Other
Composition Of Two Shm Of Same Period At Right Angles To Each Other : The composition of two simple harmonic motions (SHMs) of the same period at right angles to each other forms a Lissajous figure
Let the two motions at right angles be
x = A sin (ω t ) …(1)
y = B sin (ω t + ϕ) …(2)
along the x and y -axis respectively Equation (1) gives
Equation (2) gives
Squaring we get,
This is the equation of an ellipse.
Case 1: For Φ = 0-
∙
The equation becomes
∙
This is the equation of a straight line. Thus, the resultant motion is a SHM along a straight line, passing through the origin, inclined at an angle
Case 2: For Φ =
∙
The equation becomes
∙
The resultant SHM is along a straight line inclined at
Case 3: For Φ =/2
∙
The equation becomes
If A = B , the equation is x 2 + y 2 = A 2 , which is a circle.
Case 4: For Φ =/4
∙
The equation becomes
∙ which is the equation of an oblique ellipse.
Special Cases Of Simple Harmonic Motions
1.
If
2.
If
3.
If
Then x 2 + y 2 = A 2 (Equation of circle)
The above figures are called Lissajous figures.
∙ Suppose two forces act on a particle, the first alone would produce a simple harmonic motion in x -direction given by
x = A 1 sin ω t …(i)
and the second would produce a simple harmonic motion in y -direction given by
y = A 2 sin (ω t + δ) …(ii)
∙ The amplitudes A 1 and A 2 may be different and their phases differ by δ. The frequencies of the two simple harmonic motions are assumed to be equal. The resultant motion of the particle is a combination of the two simple harmonic motions. The position of the particle at time t is ( x , y ) where x is given by equation (i) and y is given by (ii). The motion is thus two-dimensional and the path of the particle is in general an ellipse. The equation of the path may be obtained by eliminating t from (i) and (ii).
By (i),
Thus,
Putting in (ii)
y = A 2 [sin ω t cos δ + cos ω t sin δ]
or,
or,
∙ This is an equation of an ellipse and hence the particle moves in ellipse. Equation (i) shows that x remains between – A 1 and + A 1 and (ii) shows that y remains between A 2 and – A 2 . Thus, the particle always remains inside the rectangle defined by
∙ The ellipse given by (12.29) is traced inside this rectangle and touches it on all the four sides (figure).
(a)
The two simple harmonic motions are in phase. When the x -coordinate of the particle crosses the value 0, the y -coordinate also crosses the value 0. When x -coordinate reaches its maximum value A 1 , the y -coordinate also reaches its maximum value A 2 . Similarly, when x -coordinate reaches its minimum value – A 1 , the y -coordinate reaches its minimum value – A 2 .
If we substitute δ = 0 in equation (12.29) we get
or,
…(iii)
Simple Harmonic Motion In Physics FAQs
Q.1 : When does resulting wave in SHM has greatest amplitude?
Q.2 : When does resulting wave in SHM has minimum amplitude?
Q.3 : What is Simple harmonic motion?
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