Question

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\frac{4\pi M\Omega }{h}=10^{24}{m}^{-2}$ with h as Planck’s constant. N photons of wavelength $\lambda =8\pi \times 10^{-6}$ m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 \mu m. If the value of N is $x\times 10^{12}$, then the value of x is _____.
[Consider the spring as massless]

Answer 1

Answer: 1

Let momentum of one photon is p and after reflection velocity of the mirror is v. conservation of linear momentum
$Np\hat{i}=-Np\hat{i}+mv\hat{i}$
$mv\hat{i}=2pN\hat{i}$
$mv=2Np....\left(1\right)$
since v is velocity of mirror (spring mass system) at mean position,
$v=A\Omega$
Where A is maximum deflection of mirror from mean position. (A = 1 μm) and $\Omega$ is angular frequency of mirror spring system,
momentum of 1 photon, $p=\frac{h}{\lambda }$
$mv=2Np.....\left(i\right)$
$mA\Omega =2N\frac{h}{\lambda }$
$N=\frac{m\Omega }{h}\times \frac{\lambda A}{2}$
given, $\frac{m\Omega }{h}=\frac{10^{24}}{4\pi }{m}^{-2}$
$\lambda =8\pi \times 10^{-6}m$
$N=\frac{10^{24}}{4\pi }\times \frac{8\pi \times 10^{-6}\times 10^{-6}}{2}$
$N=10^{12}=x\times 10^{12}$