A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that 4 π M Ω h = 10 24 m - 2 with h as Planck's constant. N photons of wavelength λ = 8 π × 10 - 6 m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 μ m . If the value of N is x × 10 12 , then the value of x is________. [Consider the spring as massless]

Step-by-step solution

Total Momentum transferred on mirror = 2 N h λ (for perfect reflection) In an S H M V m a x = A . Ω Where A = amplitude and Ω = angular frequency ⇒ 2 N h λ = M V m e a n p o s i t i o n V m e a n p o s i t i o n = Ω A w h e r e A = 1 μ m ⇒ 2 N h λ = M Ω A w h e r e λ = 8 π × 10 - 6 ∴ N = λ M Ω A 2 h ⇒ N = M Ω 10 - 6 λ 2 h = M Ω 8 π × 10 - 6 × 10 - 6 2 h N = 4 π M Ω h × 10 - 12 = 10 24 × 10 - 12 N = 1 × 10 12 x = 1