Calculating Gravitational Force and Work on a Spacecraft
Problem Q2: Spacecraft Dynamics in the Earth-Moon System
At a certain instant, the Earth, the Moon, and a stationary 1250-kg spacecraft lie on the vertices of an equilateral triangle whose sides are 3.84 x 105 km in length.
Given Parameters
- Mass of the Earth ($M_E$): 5.97 x 1024 kg
- Mass of the Moon ($M_M$): 7.35 x 1022 kg
- Total Mass ($M_{Total}$): 6.04 x 1024 kg (Verification required)
The simplest way to approach this problem is to find the gravitational force between the spacecraft and the center of mass (CM) of the Earth-Moon system, which is:
…from the center of the Earth. Your group should calculate the center of mass to verify this value. Once verified, use this mass value to calculate the following:
Required Calculations
- Gravitational Force: Find the magnitude and direction of the total gravitational force acting on the spacecraft by the Earth and the Moon. Specify the direction and angle as a line from the spacecraft to the Earth.
Minimum Work: What is the minimum amount of work required to move the spacecraft to a point infinitely far from the Earth and Moon? Ignore gravitational effects of other planets, the Sun, and other moons. Recall the work formula: $W = $
; where $m$ is the mass of the spacecraft and $M$ is the Earth-Moon mass. Explain why the work done by gravity is negative.
Solution Steps
1. Calculating the Center of Mass (CM)
First, we calculate the center of mass by placing our coordinate system on the Earth. The mass of the Earth is: 5.97 x 1024 kg, and the mass of the Moon is 7.35 x 1022 kg. The total mass of the Earth-Moon system is: 6.04 x 1024 kg.
CM Coordinates for the Earth-Moon System
The x-center and y-center of mass are calculated as follows:
Since we are finding the center of mass of the Earth-Moon system, and both bodies lie on the x-axis (not including the spacecraft, which has a y-component), the y-center of mass of the Earth-Moon system is:
2. Calculating Gravitational Force
We can now use this center of mass result by treating the Earth-Moon system as a single mass located on the x-axis at 4.67 x 106 meters. Then, using the Law of Gravitation, we can more easily calculate the force on the spacecraft to be: 3.4 N, at an angle of 0.61 degrees from the Earth-spacecraft line. Let’s demonstrate this:
Distance to the Center of Mass
The distance from the spacecraft to the center of mass of the Earth-Moon system is shown to be:
…making an angle of 0.69 degrees, as shown above.
To derive this value, we use the inner, larger triangle, which has a base of:
With the new base and distance to the spacecraft, we can easily compute the new hypotenuse using the Pythagorean theorem:
Determining the Angle
The previous triangle was equilateral, hence all angles were 60 degrees. By computing the angles of the new triangle, we can subtract the new angle from the old angle to get the 0.69 degrees difference. We use the sine function to determine the top angle for the new triangle, formed by treating the Earth and Moon as a single mass (the center of mass):
Subtracting this value from half of 60 degrees yields:
3. Calculating Minimum Work Required (W)
What is the minimum amount of work that must be done to move the spacecraft to a point far from Earth and Moon? (Ignoring other celestial bodies.)
Recall the work formula: $W = $
; where $m$ is the mass of the spacecraft and $M$ is the Earth-Moon mass.
Why is Work Done by Gravity Negative?
The work done by gravity is negative because gravity is an attractive force. To move the spacecraft away from the Earth-Moon system (increasing its potential energy), an external force must do positive work against the gravitational field. Since the gravitational force acts in the opposite direction of the displacement (outward), the work done by gravity is negative.
Assuming your external force is equivalent to the gravitational force but opposite in sign as we move the spacecraft toward infinity, the required work is: