Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies `omega_(1)` and `omega_(2)` and have total energies `E_(1)` and `E_(2)` repsectively. The variations of their momenta `p` with positions `x` are shown in figures. If `(a)/(b) = n^(2)` and `(a)/(R) = n`, then the correc t equation(s) is (are) :

Options

A. `E_(1)omega_(1)=E_(2)omega_(2)`
B. `(omega_(2))/(omega_(1))=n^(2)`
C. `omega_(1)omega_(2)=n^(2)`
D. `(E_(1))/(omega_(1))=(E_(2))/(omega_(2))`

Answer

BD

Explanation

`x=Asin(omegat)`
`V=Acos(omegat)`
At mean position
`P_(1)=mA_(1)omega_(1)=b`,`A_(1)=a`(from ellipse)
`P_(2)=mA_(2)omega_(2)=R`,`A_(2)=R` (From circle)
Also, `(a)/(b)=n^(2) and (a)/(R)=nimplies (R)/(b)=n` , `E_(1)=(P_(1)^(2))/(2m)` and `E_(2)=(P_(2)^(2))/(2m)`
`(E_(1))/(E_(2))=((b)/(R))^(2)=(1)/(n^(2))implies(momega_(1)^(2)A_(1)^(2))/(momega_(2)^(2)A_(2)^(2))=(1)/(n^(2))` `implies (omega_(1))/(omega_(2))=(1)/(n^(2))implies (E_(1))/(omega_(1))=(E_(2))/(omega_(2))`