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Question:

A perfect reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that 4πMΩh = 1024m⁻²; with h as Planck's constant. N photons of wavelength λ = 8π × 10⁻⁶ m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 μm. If the value of N is X × 10¹², then the value of x is ____ . [Consider the spring as massless]

1

Solution:

Correct option is A. 1
Momentum transferred on mirror = 2Nh/λ
2Nh/λ = MV (mean position)
V (mean position) = ΩA (where A = 1 μm)
2Nh/λ = MΩA
(where λ = 8π × 10⁻⁶)
N = MΩ(10⁻⁶)λ/2h = MΩ8π × 10⁻⁶ × 10⁻⁶/2h
N = 4πMΩh × 10⁻¹²/2 = 1024 × 10⁻¹²/2
N = 1 × 10¹²
x = 1