Question regarding Simple Harmonic Motion

[sankalpmittal]

Homework Statement

The two linear simple harmonic motions of equal amplitudes , and angular frequencies ω and 2ω are imposed on a particle along the axes of X and Y respectively. If the initial phase difference between them is π/2 , then find the resultant path followed by the particle.

Homework Equations

http://en.wikipedia.org/wiki/Simple_harmonic_motion

The Attempt at a Solution

I tried solving question using x= a sinωt
And other is y=a sin(2ωt + π/2)
Solving them , I get the wrong answer in terms of x and y...

Please help !

Thanks in advance...

 

[gneill]

Can you show how you went about "solving them"? We need to see your attempt if we're to know how to help.

 

[ehild]

Hi sankalpmittal,

Show please what you have tried?
x=sin(ωt) and y=sin(2ωt+π/2) =cos(2ωt) is the parametric representation of a Lissajous curve. See http://en.wikipedia.org/wiki/Lissajous_curve

Try to eliminate the ωt term.

ehild

 

[sankalpmittal]

Hi sankalpmittal,

Show please what you have tried?
x=sin(ωt) and y=sin(2ωt+π/2) =cos(2ωt) is the parametric representation of a Lissajous curve. See http://en.wikipedia.org/wiki/Lissajous_curve

Try to eliminate the ωt term.

ehild

Firstly I solved this question using :
a : Amplitude
x=a cosωt
y=a sin(2ωt+π/2)
y=a cos (2ωt)
y= a( cos²ωt-sin²ωt)
y= a(2cos²ωt-1)
y= a{(2x²/a²)-1)

I noticed that this does not match the answer given in my textbook...

Then I made a second attempt :
x=a sinωt
y=a sin (2ωt+π/2)
y=a cos(2ωt)
y= a(1-2sin²ωt)
y= a{1-(2x²/a²)}

Again this does also not match with the answer in my textbook...

Then I made the third attempt :

x= a cosωt
y= a cos(2ωt+π/2)
y= -asin(2ωt)
y= -2a sin(ωt)cos(ωt)
y=-2a √(1-sin²ωt) (x/a)
y=-2a √{1-(x²/a²)} (x/a)
On squaring both sides and simplifying , I got :

y= 4x²{1-(x²/a²)}

This answer matched with the answer given in my textbook...

I just wanted to know , where I went wrong in my first and second attempt...

 

[ehild]

Firstly I solved this question using :
a : Amplitude
x=a cosωt
y=a sin(2ωt+π/2)
y=a cos (2ωt)

They are ins phase, instead of being shifted by pi/2.

Then I made a second attempt :
x=a sinωt
y=a sin (2ωt+π/2)
y=a cos(2ωt)
y= a(1-2sin²ωt)
y= a{1-(2x²/a²)}

Again this does also not match with the answer in my textbook...

.

That is correct, if x and y are as you assumed.

Then I made the third attempt :

x= a cosωt
y= a cos(2ωt+π/2)
y= -asin(2ωt)
y= -2a sin(ωt)cos(ωt)
y=-2a √(1-sin²ωt) (x/a)
y=-2a √{1-(x²/a²)} (x/a)
On squaring both sides and simplifying , I got :

y= 4x²{1-(x²/a²)}

This answer matched with the answer given in my textbook...

But this answer is wrong as y is not squared. It was correct before squaring.

If you add two perpendicular vibrations you can have different y(x) curves, depending on the initial phases.


ehild

 

[sankalpmittal]

They are ins phase, instead of being shifted by pi/2.

.

That is correct, if x and y are as you assumed.



But this answer is wrong as y is not squared. It was correct before squaring.

If you add two perpendicular vibrations you can have different y(x) curves, depending on the initial phases.


ehild

OK , so there was a typo. The answer I got in my third attempt was y²=4x²{1-(x²/a²)} , which matched with the answer given in my textbook. But unfortunately the answer which I got in my first and second attempt , did not match with that given in my textbook.

According to you , both the equations , in my second and third attempt are correct , yet they are different and only latter match with the answer in my textbook. How ?

Also what do you mean by "in phase and not shifted by pi/2" in my first attempt ? Can you explain a bit comprehensively ?

 

[ehild]

Your first attempt was
x=a cosωt
y=a sin(2ωt+π/2) which is equivalent to
y=a cos (2ωt)I do not see any initial phase difference between x and y.
Your second attempt is true and so is the third one. Phase difference between x and y is not a clear concept. You get x,y curves of different form if x=sin(ωt), y=cos(2ωt) (y=1-2x²) and when x=cos(ωt) and y=sin(2ωt) (y²=4x²(1-x²))

ehild

 

[SammyS]

Homework Statement

The two linear simple harmonic motions of equal amplitudes , and angular frequencies ω and 2ω are imposed on a particle along the axes of X and Y respectively. If the initial phase difference between them is π/2 , then find the resultant path followed by the particle.

Homework Equations

http://en.wikipedia.org/wiki/Simple_harmonic_motion
...

A better link to consider might be: http://en.wikipedia.org/wiki/Lisajous .

Also see: http://en.wikipedia.org/wiki/Lemniscate_of_Gerono .