It has been proposed that a spaceship might be propelled in...

It has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil. How large must the sail be if the radiation force is to be equal in magnitude in the Sun's gravitational attraction? Assume that the mass of the ship $+$ sail is $1500 \mathrm{~kg}$, that the sail is perfectly reflecting, and that the sail is oriented perpendicular to the Sun's rays. See Appendix $C$ for needed data. (With a larger sail, the ship is continually driven away from the Sun.)

It has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil. How large must the sail be if the radiation force is to be equal in magnitude in the Sun's gravitational attraction? Assume that the mass of the ship $+$ sail is $1500 \mathrm{~kg}$, that the
sail is perfectly reflecting, and that the sail is oriented perpendicular to the Sun's rays. See Appendix $C$ for needed data. (With a larger sail, the ship is continually driven away from the Sun.)

Key Concepts

Force Equilibrium

This concept pertains to the balance between opposing forces. In the context of solar sails, it involves equating the force due to radiation pressure against the gravitational force, determining the necessary sail area required for propulsion or station-keeping.

Photon Momentum Transfer

When photons strike a surface, they impart momentum to it. In a perfectly reflecting surface, this transfer is effectively doubled, enhancing the force exerted by the radiation pressure. This principle is essential in calculating the efficiency and required size of a solar sail for a given spacecraft.

Gravitational Force

This is the attractive force between any two masses as described by Newton’s law of universal gravitation. For a spacecraft in the solar system, the Sun's gravitational pull is the dominant force that must be countered or balanced by other forces such as radiation pressure.

Solar Sail Propulsion

This involves using a large, lightweight, and reflective surface to harness the momentum transfer from solar photons to generate thrust. The design and operation of solar sails depend on the efficient conversion of radiation pressure into a propulsive force, making mass and area key parameters.

Radiation Pressure

This concept refers to the pressure exerted by electromagnetic radiation, such as sunlight, on a surface. The force arises from the transfer of momentum from photons to the surface upon interaction, and it is fundamental to understanding how solar sails can propel spacecraft.

Transcript

00:01 So this question asks us to determine the size of a solar sail needed so that the force of attraction due to gravity on the spaceship due to the sun is equal to the force due to radiation pressure from the solar sail.

00:17 And we're told that the ship is of mass 1 ,500 kilograms and that the sail is perfectly reflecting and the sail is orientated perpendicular to the sun's rays.

00:28 So all of the data that we need for this question i have written down here.

00:32 So we're going to need the acceleration due to gravity, or the newton's gravitational constant g, which is 6 .67 by 10 to the minus 11 newton meters per kilogram squared.

00:42 We're going to need the mass of the sun, m, capital m, which is equal 1 .98 by 10 to the 30 kilograms.

00:48 We have the mass of the spaceship, which is 1 ,500 kilograms, the speed of light, which is 3 by 10 to 8 meters per second.

00:55 And this is the one extra piece that we needed.

00:58 We needed the power output of the sun, which is equal to 3 .828 by 10 to the 26 watts.

01:07 So this question is fundamentally asking us.

01:11 When is, for what area of say it is the force of attraction due to gravity on the spaceship and is equal to the radiation pressure force? so we have that the force due to gravity is equal to the, radiation pressure force f4 so we know that the force due to gravity it's just uses newton's gravitational loss that's gmm over the radius squared and we're not told what radius so we're just going to leave that blank for the minute and we know that this is equal to then the radiation force which is equal to so we know that the that for a reflecting body it's equal to two times the intensity of the light source over the speed of light and we know that the know that we can break that we can write the intent on this by the area then as well since we're looking for force so this is so let's write this out again so this radiation force equal to the radiation pressure times the area since force is equal to since pressures you get the force per unit area so we're just rearranging that so then this is then equal to so we know that the radiation pressure for reflecting surface is equal to two times the intensity over the speed of light by the area...

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