Question

A perfect reflecting mirror of mass M mounted on a spring constitudes a spring-mass system of angular frequency $$\Omega$$ such that $$\dfrac{4 \pi M \Omega}{h} = 10^{24} m^{-2}$$ 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

Correct option is A. 1
Momentum transferred on mirror = $$\dfrac{2Nh}{\lambda}$$
$$\dfrac{2Nh}{\lambda} = MV_{(mean \, position)}$$
$$V_{(mean \, position)} = \Omega A$$ (where $$A = 1 \, \mu M$$)
$$\dfrac{2Nh}{\lambda} = M \Omega A$$ (where $$\lambda = 8 \pi \times 10^{-6}$$)
$$N = \dfrac{M \Omega (10^{-6}) \lambda}{2h} = \dfrac{M \Omega 8 \pi \times 10^{-6} \times 10^{-6}}{2h}$$
$$N = \dfrac{4 \pi M \Omega}{h} \times 10^{-12}$$
$$= 10^{24} \times 10^{-12}$$
$$N = 1 \times 10^{12}$$
$$x = 1$$