1for more clarification refer to hcv vector method of combining shms
The trajectory of a body with 2 simultaneous,perpendicular oscillations is determined by
x=a sin(pωt)
y=b sin(qωt+φ)
where x and y are projections of body displacement on X and Y axis.
For simpicity let us assume pω=qω=ωo.Then the equation of trajectory will be y2b2+x2a2-2xyabcosφ=sin2φ
which is general eqn of ellipse
But if qω≠pω and p≠q then the graph of trajectory on XY plane is eitehr a closed curve (whose loop number is defined by ratio n=p/q) or an open curve
[At the point where curve represents same trajectory,the velocities along X axis and Y axis become equal to zero simultaneously.The body moving with along hte curve stops exactyl at this moment at a certain point,and then moves backeward]
Q1 if p=2,q=1,φ=0,the what is path of trajectory?
Q2 If φ=π/2,and p/q=1,then what is nature of trajectory ?
Q3 If p=2/3,q=8/9 hten time peiod of oscilation is ?
Q4 If p=q=1 and φ=0,then find the trajectory
Lastly....for all the sirs..
can this question come in JEE ??
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5 Answers
1actually according to me this sum is more abt using trignometric identities rather than using the concept and it s next to impossible for coming in objective
first sum
x=2asinwtcoswt
y=bsinwt
solving it for coswt u can get ur eq,....
12nd one is a relatively simple sum
x=psinwt
y=pcoswt
x2+y2=p2(sin^2+cos^2)
or
x2+y2=p2
this is the eq of a circle
14)
x=psinwt
y=psinwt
therefore y=x and m=1 and c=0
this is the special case of a st line passing thru the origion....
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24In Q2
how did u write x=psinωt,y=pcosωt
it should be x=a sinωt,y=b cosωt naa ?
In Q4 too
x=psinwt
y=psinwt ??????