The work done against a force F(r) in moving an object from r = r1 to r = r2 is integral F(r) dr limits R2 at top and r1 at bottom. The gravitational attraction between two masses m1 and m2 at distance d is given be F=GMM/d2, G = 6.67 x 10-11. Find the work done in lifting a 4000 kg payload from the surface on the moon to a height of 25000m above its surface. The mass of the moon can be taken as 7.3 x 1022 and its radius at 1.7 x 106.

Question

The work done against a force F(r) in moving an object from r = r1 to r = r2 is integral F(r) dr limits R2 at top and r1 at bottom. The gravitational attraction between two masses m1 and m2 at distance d is given be F=GMM/d2, G = 6.67 x 10-11. Find the work done in lifting a 4000 kg payload from the surface on the moon to a height of 25000m above its surface. The mass of the moon can be taken as 7.3 x 1022 and its radius at 1.7 x 106.

Answer 1

The work done would be

G m1*m2[1/r1 - 1/r2]

r1 = 1.7*10^6 m
r2 = 1.7*10^6 + 2.5*10^4 = 1.725 *10^6 m

Make sure mass is in kg and use the value of G that they provided. The answer should be in Joules.

Be more careful to show units with your numbers.

Answer 2

To find the work done in lifting the payload from the surface of the moon to a height of 25000m above its surface, we can use the given gravitational force equation and integrate it over the distance moved.

First, let's calculate the distance between the surface of the moon and the desired height above it:
h = 25000m
R_moon = 1.7 x 10^6m
distance = R_moon + h

Now, let's substitute the given values into the gravitational force equation:
F = G * (m1 * m2) / d^2
Here, m1 is the payload mass and m2 is the mass of the moon.

Given:
m1 = 4000 kg
m2 = 7.3 x 10^22 kg
G = 6.67 x 10^-11 m^3/(kg * s^2)
d = distance from the center of the moon to the payload = R_moon + h

Substituting the values:
F = (6.67 x 10^-11) * (4000 * 7.3 x 10^22) / (R_moon + h)^2

Now, we need to calculate the work done against this force. The work done is given by the integral of the force over the distance moved:

Work = ∫ F(r) dr (from r1 to r2)

Here, r1 is the distance from the center of the moon to the surface, which is the moon's radius - R_moon.
r2 is the distance from the center of the moon to the desired height above its surface, which is R_moon + h.

Work = ∫ (6.67 x 10^-11) * (4000 * 7.3 x 10^22) / (r^2) dr (from R_moon to R_moon + h)

Integrating this equation will give us the work done against the gravitational force when moving the payload from the surface of the moon to a height of 25000m above it.

Note: The integration process involves advanced mathematical calculus, and the result may be a bit complicated. It is recommended to use appropriate software or a calculator capable of performing symbolic integration for precise results.