Physics3 question on light, radiation pressure

[Yroyathon]

Hi folks, first time poster. I'm a math undergrad taking my final physics3 requirement. I'm doing ok in the class, but about 2 out of 8 of the homework problems confound me. I removed the numbers below and replaced them with variables (does that matter on this site?). Mostly I'm interested in getting the concepts right, which approach to use, when to use certain equations.

Below is one problem I've tried to answer 4 times already, incorrectly.

Homework Statement

Suppose that you want to use the radiation pressure from a beam of light to suspend a piece of paper in a horizontal position; the paper has an area of A_1 cm2 and a mass of m_1 grams. Assume that there is no problem with balance, that the paper is dark and absorbs the beam fully, and that the entire beam can be used to hold the paper against the pull of gravity. How many watts must the light produce?

Homework Equations

For radiation pressure, I've used u=F/A, where F is the gravitational force, F=m_1 * (9.8). I wasn't sure if the A here should be the area of the paper A_1, or if we should consider the paper as a point mass and use unit area, A_1 = 1?... so either u=F=m_1*(9.8) or u=F/A=m_1*(9.8)/A_1.

To calculate power, I've been using Power=c*u*A, where c is the speed of light (3*10^8), u is the radiation pressure, and A is the area of the paper A_1. So Power=(3*10^8)*m_1*(9.8)*A_1.

But this hasn't worked. I tried using Power=c*u, and saying that A = 1 there, and using u = F/A = m_1*(9.8)/A_1, so that Power=c*u = (3*10^8)*m_1*(9.8)/A_1, but this didn't work either.

i'm kind of guessing at the approach, and was hoping someone here could explain the parts I'm not getting.

thanks in advance!
,Yroyathon

 

[Redbelly98]

Welcome to PF

To calculate power, I've been using Power=c*u*A, where c is the speed of light (3*10^8), u is the radiation pressure, and A is the area of the paper A_1. So Power=(3*10^8)*m_1*(9.8)*A_1.

Including A is the best approach.

What does u, the radiation pressure, need to be here?

In the above, you are equating u with m_1*(9.8 m/s^2), which is a force and therefore must be wrong because a pressure cannot equal a force.

 

[Yroyathon]

yes, you are right.

after another hour or so of thought, I manage to convince myself that the area in this problem does not matter.

u=F/A=m*g/A, and Power=u*c*A, so Power=m*g*c. Using this the answer turned out to be correct.

The way I managed to convince myself is that if the beam was spread out to cover the entire piece of paper equally, or if the beam was focused on the exact center of the paper, nothing would change in the beam's ability to push-up/levitate the paper.

Thanks a lot for reading and replying. Posting the problem here was helpful because I had to focus even harder on what I knew and didn't know so that I could write the post and describe my attempt at the problem.

 

[Redbelly98]

Glad it worked out. Good observation about the area, by the way.