Simple Harmonic Motion In Physics, Composition Of Two SHM, Important Topics For JEE 2025
Shrivastav 16 Jun, 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
- 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
=
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
- 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
to the x -axis
CASE 2: For ϕ = π
-
The equation becomes
-
The resultant SHM is along a straight line inclined at
to the x -axis
CASE 3: For ϕ =π/2
-
The equation becomes
which is an ellipse
If A = B , the equation is x 2 + y 2 = A 2 , which is a circle.
CASE 4: For ϕ =π/4
-
The equation becomes
- which is the equation of an oblique ellipse.
Special Case :
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 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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