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Perfectly reflecting mirror of mass M mo...

Perfectly reflecting mirror of mass M mounted on a spring consitute a spring mass system of angular frequency `Omega` such that `(4piM Omega)/h=10^(24) m^(-2)` where h is plank constant. N photons of wavelength `lambda=8pixx10^(-6)m ` strikes the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 `mum`. if the value of N is `x xx 10^(12)`, then find value of x.

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The correct Answer is:
[1]
Let momentum of one photon is p. It is given that mirror is perfectly reflecting, hence it gets reflected with the same momentum p along opposite direction. Let v be the velocity gained by mirror in the same direction as that of the incident photon. According to conservation of linear momentum we can write the following equation for N photons.
`Np=-Np+mv`
`implies mv = 2Np " "...(i)`
Let `A (A=1)` be the amplitude of oscillation for the mirror and `Omega` be the angular frequency of the mirror then we should know that `v = AOmega` And further linear momentum p of the photon can be written as `p=h/lamda` .
So equation (i) can be written as follows:
`impliesmAOmega=2N(h//lamda)`
`impliesN=(mOmega)/hxx(lamdaA)/2" "...(ii)`
Further it is given that `(mOmega)/h=10^(24)/(4pi) andlamda=8pixx10^(-6),`
Substituting in equation (ii),
`N=10^(24)/(4pi)xx(8pixx10^(-6)xx10^(-6))/2`
`implies N = 1 xx1^(12)`
On comparing we get x = 1.
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