Suppose you're an astronaut and you're sitting in a 6000kg space capsule riding on a 4000 m x 8000 m sail (whose mass is included in the 6000 kg). You're orbiting in the vicinity of a distant bright hot star giving off light with an intensity 8 x 10^9 W/m^2 at a distance where the gravitational Acceleration is 30 m/s^2. Find a.) the radiation pressure on your sail. B.) the force of light on your sail. C.) is it strong enough to escape the star?
To find the radiation pressure acting on your sail, you can use the formula:
Pressure = Intensity of light × Reflectivity / Speed of light
a) Radiation pressure on the sail:
Given:
Intensity of light (I) = 8 x 10^9 W/m^2
Reflectivity (R) = This is a value between 0 and 1 that represents how well the sail reflects light. Assuming a perfect reflector, R = 1.
Speed of light (c) = 3 x 10^8 m/s
Using the given values, we can calculate the radiation pressure using the formula mentioned earlier:
Pressure = (Intensity of light × Reflectivity) / Speed of light
= (8 x 10^9 W/m^2 × 1) / (3 x 10^8 m/s)
= (8/3) × 10^9 / (3 × 10^8)
= (8 / 3) × (10^9 / (3 × 10^8))
= (8 / 3) × (10^1 / 10^0)
= (8 / 3) × 10^(1 - 0)
= (8 / 3) × 10^1
= 8 / 3 × 10
≈ 2.67 × 10 N/m^2
Therefore, the radiation pressure on your sail is approximately 2.67 × 10 N/m^2.
b) Force of light on the sail:
The force exerted on the sail can be calculated using the formula:
Force = Pressure × Area
Given:
Area (A) = 4000 m × 8000 m = 32,000,000 m^2 (since the sail dimensions are given as 4000 m x 8000 m)
Using the value of pressure calculated in part (a), we can calculate the force:
Force = Pressure × Area
= 2.67 × 10 N/m^2 × 32,000,000 m^2
= 2.67 × 10 N/m^2 × 3.2 × 10^7 m^2
= 8.544 × 10^7 N
Therefore, the force of light on your sail is approximately 8.544 × 10^7 N.
c) Escape from the star:
To determine if the force of light on the sail is strong enough to escape the star, we need to compare it to the force of gravity pulling the sail towards the star.
Given:
Mass of the sail and capsule (m) = 6000 kg
Gravitational acceleration (g) = 30 m/s^2
Force of gravity (Fg) = mass × gravitational acceleration
= 6000 kg × 30 m/s^2
= 180,000 N
Comparing this to the force of light on the sail (8.544 x 10^7 N), we see that the force of gravity is significantly weaker.
Therefore, the force of light on the sail is strong enough to overcome the force of gravity and allow the sail to escape from the star.