Composition of 2 SHM in same direction

In summary: Sorry for the confusion.In summary, the problem involves finding the phase difference between two simple harmonic motions with equal amplitudes and frequencies, where the resultant amplitude is also equal to the individual motions. By using the cosine law and representing the waves as phasors, the phase difference can be calculated to be either 2π/3 or 4π/3.
  • #1
vissh
82
0
elllllo :D

Homework Statement

<q>
<Q>A particle is subjected to 2 simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the individual motions,find the phase difference between the individual motions.

Homework Equations


>Let x1 = A1sin(wt) and x2 = A2sin(wt + a)
Then, the resultant motion is also a SHM given by : x = A sin(wt +b)
where A = [(A1)2 +(A2)2 + 2(A2)(A1) cos(a) ] 1/2

The Attempt at a Solution


Let the amplitude of individual motions was "A" and the phase difference was "d".
So, A = [ A2 + A2 + 2(A)(A)cosd ] 1/2
=> A= A [ 2 + 2cosd]1/2
=> 2 + 2cosd = 1
=> cosd = -(1/2)
=> d = 2[tex]\pi[/tex]/3 or 4[tex]\pi[/tex]/3 [taking values b/w [0,2[tex]\pi[/tex]]
But the answer got only 2[tex]\pi[/tex]/3
Did i did wrong at some place ?
Thanks for reading (^.^)
 
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  • #2
Take a close look at your cosine law.
 
  • #3
Looks correct to me.

It may help you to think of these situations in terms of "phasors"
http://en.wikipedia.org/wiki/Phasor

Draw a vector representing each wave with a length equal to the amplitude, and an angle equal to the phase. The resultant vector will give you the amplitude and phase of your composition.
 
  • #4
Thanks to you both for replying ^.^
Hmm cosine law-- U mean " cosd = -(1/2) " .
Whats wrong abt it ?? O.O
Or u want to say i sud solve like : d = arccos(-1/2)
and take the principal values only which lies in [0,pie] and thus, get 2(pie)/3 only.
Is that so?
 
  • #5
Beaker87 said:
Looks correct to me.

You're right of course; I was thinking of the basic cosine law without considering where the angle "a" was coming from in this problem.
 

1) What is the composition of two SHM in the same direction?

The composition of two SHM (simple harmonic motion) in the same direction refers to the combination of two oscillatory motions that are both moving in the same direction. This means that the two motions have the same amplitude, frequency, and phase, resulting in a combined motion that follows the same path.

2) What is the resulting motion when two SHM are combined in the same direction?

The resulting motion when two SHM are combined in the same direction is a new oscillatory motion with a larger amplitude and the same frequency and phase as the individual motions. This means that the combined motion will have a greater displacement from the equilibrium position and will follow the same path as the individual motions.

3) How does the amplitude of the combined motion compare to the individual motions?

The amplitude of the combined motion is equal to the sum of the amplitudes of the individual motions. This is because when two SHM are combined in the same direction, their amplitudes add together to create a larger amplitude in the combined motion.

4) What is the difference between constructive and destructive interference in the composition of two SHM in the same direction?

In the composition of two SHM in the same direction, constructive interference occurs when the two motions have the same amplitude, frequency, and phase, resulting in a larger combined motion. Destructive interference occurs when the two motions have opposite amplitudes, resulting in a cancellation of the combined motion.

5) Can the composition of two SHM in the same direction result in a stationary motion?

No, the composition of two SHM in the same direction will always result in a combined motion with a nonzero amplitude. This is because for the two motions to cancel each other out and result in a stationary motion, they would need to have opposite amplitudes, which is not possible if they have the same frequency and phase.

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