Radiation Pressure — Question 5

Question 5

Tilting a reflecting sheet changes the vertical support force cubically

A perfectly reflecting sheet has area A and mass per area sigma. A wide beam travels vertically upward with intensity I. The sheet normal makes angle theta with vertical. A lateral constraint prevents sideways motion without providing vertical force.

1. Derive the vertical optical force and intensity required to balance weight.

2. Find the intensity multiplier when the sheet is tilted from zero to 60 degrees.

Question 5 - Solution

Tilting a reflecting sheet changes the vertical support force cubically

(a) Method and calculation

$$P_{\mathrm{hit}}=IA\cos\theta,\quad F_n=2IA\cos^2\theta/c.$$$$F_z=F_n\cos\theta=2IA\cos^3\theta/c.$$

(b) Interpretation and further calculation

$$F_z=\sigma Ag\Rightarrow\boxed{I_{\mathrm{balance}}=\frac{\sigma gc}{2\cos^3\theta}}.$$$$\boxed{I(60^\circ)/I(0)=\sec^3 60^\circ=8}.$$

Physical check

This is a force-balance condition, not a claim of stable levitation; the horizontal reaction is supplied by the stated constraint.