Physics – what is the maximum mass of the sphere that can be supported by the laser beam?
i’m stumped on this hw question can anyone help me out. the full question goes:
A laser produces a power of 4.0 W in a beam 0.40mm in diameter. if the laser bean is aimed upward to levitate a 20 ohms m diameter sphere, what is the maximum mass of the sphere that can be supported by the laser beam?
Id really appreciate if any one knows.
Thanks a bunch ahead of time
The laser is made of photons, and a photon has a momentum. When a photon hits something, and gets absorbed, the momentum gets imparted to the object (conservation of momentum). If the photon gets reflected back, its momentum changes to negative its original value, so the object needs to pick up twice the momentum to conserve momentum. Let p stand for momentum of light and P stand for momentum of sphere, and i stands for initial and f stands for final:
pi = pf +Pf = -pi + Pf ==> Pf = 2pi
That is the general principle. So the sphere will be held up best if it is reflective, not black. Now lets answer the question. We need to know the momentum of a photon, and the number of photons, to find the force on the sphere. Einstien will tell us the momentum of a photon is given by
E=pc
where E is energy and c is the speed of light. Planck will tell us that the energy of a photon is given by Planck’s constant h and the frequency f:
E=h f
So the momentum is p=hf/c for a single photon. Now how many photons are in your laser? Obviously, you will find the photons per second by dividing the total power (energy per second) by the energy in a single photon:
photons/sec = Power/hf
So how much momentum is contained in a seconds worth of light?
momentum/sec = photon/sec * momentum/photon
= Power/hf * hf/c = Power/c
So how much momentum per second does the sphere get when the laser bounces off of it? Recall the factor of 2:
momentum of sphere/sec = 2 Power/c
Now a change in momentum over time is a force (derivative of P is force), so the momentum/sec is actually a force pushing the ball upward.
Force due to laser = 2 Power/c
Balance this with the gravitational force to find the max weight supported by the laser:
2 Power/c = mass * g
mass=2 Power/gc
where g=9.8m/s^2, c=3*10^8 m/s, and Power = 4.0W
I’ll let you plug in numbers. Note that I did not use the diameter of the sphere or the beam. If the diameter of the beam was bigger than the sphere, only some of the light would bounce off. If the diameter of the beam were smaller but still on the same scale as the diameter of the sphere, the outside part of the beam would reflect off of the curved sphere in a direction other than backwards, and the momentum transfered would not be as great, so the force due to the laser would be less. But the beam is sooo much smaller than the sphere, that the sphere will look flat and you can ignore the diameters. If you wanted to solve this exactly, taking into account the curvature of the sphere, you would have to integrate. But the beam is so small compared to the sphere that it would hardly change the answer.
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