During 2004, japanese scientists successfully tested two solar sails. One had somewhat complicated shape that we shall model as a disk 9 m in diameter and 7.5 um thick. The intensity of solar energy at this location was about 1400 W/m
^2. (a) What force did the sun's light exert on this sail, assuming that it struck perpendicular to the sail and that the sail was perfectly reflecting? (b) If the sail was made of magnesium, of density 1.74 g/cm^3, what acceleration would the sun's radiation give to the sail? (c) Does the acceleration seem large enough to be feasible for space flight? In what ways could the sail be modified to increase its acceleration?
If perfectly reflecting the radiation pressure is 2 I/c
p = 2 *1400 W/m /3*10^8 m/s
= 9.4 *10^-6 Pascals
That is like 10^-10 atmospheres, like tiny
multiply the p by pi (D/2)^2 to get force
calculate mass from density and do a = F/m
To find the force exerted by the sun's light on the solar sail, we need to calculate the pressure of the light on the sail. The pressure is given by the formula:
Pressure = Intensity of light * Area
(a) The area of the sail can be calculated using the formula for the area of a disk:
Area = π * (radius)^2
The radius of the disk is half the diameter, so:
radius = 9 m / 2 = 4.5 m
Substituting the values into the formula, we get:
Area = π * (4.5 m)^2 = 63.62 m^2
Now, we can calculate the force using the pressure formula. The intensity of light is given as 1400 W/m^2, so:
Force = Pressure = Intensity of light * Area = 1400 W/m^2 * 63.62 m^2
Calculating the force:
Force = 1400 W/m^2 * 63.62 m^2 = 89,068 N
Therefore, the force exerted by the sun's light on the sail is approximately 89,068 N.
(b) To find the acceleration of the sail, we first need to calculate its mass. The mass can be found using the formula:
Mass = Volume * Density
The volume of the sail can be calculated using the formula for the volume of a cylinder:
Volume = π * (radius)^2 * thickness
Substituting the values into the formula, we get:
Volume = π * (4.5 m)^2 * 7.5 um
Note: um stands for micrometer, which is equal to 1 x 10^-6 meters.
Converting micrometers to meters:
7.5 um = 7.5 x 10^-6 m
Substituting the values into the formula, we get:
Volume = π * (4.5 m)^2 * 7.5 x 10^-6 m
Calculating the volume:
Volume ≈ 0.000477 m^3
Now, we can calculate the mass using the given density of magnesium:
Density = 1.74 g/cm^3
Converting grams to kilograms:
Density = 1.74 x 10^3 kg/m^3
Calculating the mass:
Mass = Volume * Density ≈ 0.000477 m^3 * 1.74 x 10^3 kg/m^3 ≈ 0.829 kg
To find the acceleration, we can use Newton's second law of motion:
Force = Mass * Acceleration
Rearranging the equation, we get:
Acceleration = Force / Mass = 89,068 N / 0.829 kg
Calculating the acceleration:
Acceleration ≈ 107,396 m/s^2
Therefore, the sun's radiation would give the sail an acceleration of approximately 107,396 m/s^2.
(c) The acceleration seems extremely large for space flight. It is much higher than any practical acceleration that can be sustained by a spacecraft with astronauts or delicate equipment. In addition, the force of the sun's light is not constant, as it depends on the intensity of light at a given location in space.
To increase the acceleration of the sail, there are a few modifications that can be made:
1. Reducing the mass: Using lighter materials for the sail can decrease its mass and increase the acceleration for a given force.
2. Increasing the intensity of light: Positioning the sail closer to the sun or using reflective surfaces can increase the intensity of light, thereby increasing the force and acceleration.
3. Increasing the surface area: By increasing the surface area of the sail, more light can be captured and the force and acceleration can be increased.
Keep in mind that these modifications may come with their own limitations and challenges, such as structural integrity, heat management, and maintaining the orientation of the sail towards the sun.