Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant): (a) sin ωt – cos ωt (b) sin³ ωt (c) 3 cos (p/4 – 2ωt) (d) cos ωt + cos ³ωt + cos ⁵ωt (e) exp (–ω²t²) (f) 1 + ωt + ω²t²
a)
The given function is sin ω t − cos ω t .
The general equation of Simple Harmonic Motion (SHM) is given by,
y = a sin ( ω t + ϕ ) (1)
Let f ( x ) be the given function.
f ( x ) = sin ω t − cos ω t
Rearranging the terms of above expression by multiplying and dividing by 2 , we get
f ( x ) = 2 ( 1 2 sin ω t − 1 2 cos ω t ) = 2 ( sin ω t × cos π 4 − cos ω t sin π 4 ) = 2 sin ( ω t − π 4 ) (2)
After compare equation (2) with equation (1), we find that the given function represent a simple harmonic motion.
The time period of the periodic motion is,
T = 2 π ω
Thus, the function sin ω t − cos ω t represents SHM.
b)
The given function is sin 3 ω t .
Let f ( x ) be the given function.
f ( x ) = sin 3 ω t
Using the trigonometry, we get
f ( x ) = 1 4 ( 3 sin ω t − sin 3 ω t ) = 1 4 ( 3 sin ( ω t + 0 × ϕ ) − sin 3 ( ω t + 0 × ϕ ) ) (3)
After compare the equation (1) and equation (3), we find that each individual function represents a SHM but the superposition of both the waves is not in SHM. The resulting wave is periodic in nature but not in SHM.
Thus, the function sin 3 ω t is not in SHM.
c)
The given function is 3 cos ( π 4 − 2 ω t )
The general equation of Simple Harmonic Motion (SHM) is given by,
y = a cos ( ω t + ϕ ) (4)
Let f ( x ) be the given function.
f ( x ) = 3 cos ( π 4 − 2 ω t )
According to trigonometric identity,
cos ( − x ) = cos ( x )
Rearranging the terms of above expression, we get
f ( x ) = 3 cos ( 2 ω t − π 4 ) (5)
After compare the equation (4) and equation (5), we conclude that the given function is represents SHM with period given by,
T = 2 π ϖ
Thus, the function 3 cos ( π 4 − 2 ω t ) is in SHM.
d)
The given function is cos ( ω t ) + cos ( 3 ω t ) + cos ( 5 ω t )
Let f ( x ) be the given function.
f ( x ) = cos ( ω t ) + cos ( 3 ω t ) + cos ( 5 ω t ) = cos ( ω t + 0 × ϕ ) + cos ( 3 ω t + 0 × ϕ ) + cos ( 5 ω t + 0 × ϕ ) (6)
After compare the equation (4) and equation (6), we conclude that each individual function represents a SHM but the superposition of both the waves is not in SHM. The resulting wave is periodic in nature but not in SHM.
Thus, the function cos ( ω t ) + cos ( 3 ω t ) + cos ( 5 ω t ) is not in SHM.
e)
The given function is exp ( − ω 2 t 2 ) .
Let f ( x ) be the given function.
f ( x ) = exp ( − ω 2 t 2 )
The given function is an exponential function. The exponential functions are not repeating motion about a fixed point. As they are not periodic in nature, they are also not in SHM. All SHM are periodic in nature but not vice versa.
Thus, the function exp ( − ω 2 t 2 ) is not in SHM.
f)
The given function is 1 + ω t + ω 2 t 2 .
Let f ( x ) be the given function.
f ( x ) = 1 + ω t + ω 2 t 2
The given function is a quadratic equation. The quadratic equation is not repeating motion about a fixed point. As they are not periodic in nature, they are also not in SHM.
Thus, the function 1 + ω t + ω 2 t 2 is not in SHM.
a)
The given function is
The general equation of Simple Harmonic Motion (SHM) is given by,
Let
Rearranging the terms of above expression by multiplying and dividing by
After compare equation (2) with equation (1), we find that the given function represent a simple harmonic motion.
The time period of the periodic motion is,
Thus, the function
b)
The given function is
Let
Using the trigonometry, we get
After compare the equation (1) and equation (3), we find that each individual function represents a SHM but the superposition of both the waves is not in SHM. The resulting wave is periodic in nature but not in SHM.
Thus, the function
c)
The given function is
The general equation of Simple Harmonic Motion (SHM) is given by,
Let
According to trigonometric identity,
Rearranging the terms of above expression, we get
After compare the equation (4) and equation (5), we conclude that the given function is represents SHM with period given by,
Thus, the function
d)
The given function is
Let
After compare the equation (4) and equation (6), we conclude that each individual function represents a SHM but the superposition of both the waves is not in SHM. The resulting wave is periodic in nature but not in SHM.
Thus, the function
e)
The given function is
Let
The given function is an exponential function. The exponential functions are not repeating motion about a fixed point. As they are not periodic in nature, they are also not in SHM. All SHM are periodic in nature but not vice versa.
Thus, the function
f)
The given function is
Let
The given function is a quadratic equation. The quadratic equation is not repeating motion about a fixed point. As they are not periodic in nature, they are also not in SHM.
Thus, the function
