In a science fiction story, a major character pilots his own spacecraft, which is propelled by a stellar sail. As shown in the figure (not drawn to scale) a stellar sail is a large reflecting sheet attached to a spacecraft, much like an ordinary cloth sail is attached to a boat. Light from the star reflects off the sail, exerting a force on the sail. In the story, the spacecraft is small, and the single passenger sits in a small capsule. The trip is a short one, and there are no facilities in the capsule for rest, exercise, or nourishment. So we estimate the mass of the spacecraft, including the passenger and the stellar sail, to be m = 5000 kg, we further estimate the area of the stellar sail to be A - 28000 m2. Say that the mass of the star is M-16 MO and its luminosity is L 24600 LO, where the subscript o stands for the Sun. The mass of the Sun and its luminosity are 1.99 x 1030 kg and 3.84 x 1026 W, respectively. Find an approximate expression for the net force on the spacecraft. As a last step, determine whether the acceleration is toward or away from the star.
Part 5 of 6- Solve (Cont.) Combine the expressions for the gravitational and radiation forces to find an expression for the net force. Express your answer using the symbols defined in this tutorial. (The mass of the star is M. The mass of the spacecraft is m. The area of the sail is A. The star's luminosity is L. The distance between the center of the star and the spacecraft is r.) In the next step, you will substitute numbers to find the sign and direction of the net force. (Use any variable or symbol stated above along with the following as necessary: c and G.)
\vec{F}_{net} = \lgroup\frac{LA}{2\pic-GMm}\rgroup \frac{1}{r^2} \lgroup\frac{LA}{2\pic-GMm}\rgroup \frac{1}{r^2} \hat{r}