A 54100 kg space ship has completed two thirds of its journey from the earth to the Moon. At this position, what magnitude force would the ship's engines have to supply to be the equilibrant? Assume the mass of the moon is 7.35E + 22 kg and the distance between the earth and the moon is 384000 km.
To find the magnitude of the force that the ship's engines would have to supply to be the equilibrant, we need to balance the gravitational force between the Earth and the Moon with the force exerted by the ship's engines.
Step 1: Calculate the distance covered by the spaceship.
The spaceship has completed two-thirds of its journey from the Earth to the Moon. Since the distance between the Earth and the Moon is 384,000 km, we can calculate the distance covered by the spaceship as follows:
Distance covered = (2/3) * 384,000 km
Distance covered = 256,000 km
Step 2: Convert the distance to meters.
To maintain consistency in units, we need to convert the distance from kilometers to meters.
Distance covered = 256,000 km * 1000 m/km
Distance covered = 256,000,000 m
Step 3: Calculate the force exerted by the Earth on the spaceship.
The force exerted by the Earth on the spaceship can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where:
F is the force between two objects
G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 is the mass of the Earth (5.97 × 10^24 kg)
m2 is the mass of the spaceship (54,100 kg)
r is the distance between the center of the Earth and the spaceship (taking into account the distance already covered)
Force (Earth on spaceship) = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.97 × 10^24 kg * 54,100 kg) / (256,000,000 m)^2
Step 4: Calculate the force exerted by the Moon on the spaceship.
Similar to the calculation of the force exerted by the Earth on the spaceship, we will calculate the force exerted by the Moon on the spaceship.
Force (Moon on spaceship) = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (7.35 × 10^22 kg * 54,100 kg) / (384,000,000 m)^2
Step 5: Calculate the equilibrant force.
The equilibrant force is the force that balances the gravitational forces exerted by the Earth and the Moon on the spaceship. It can be calculated by subtracting the force exerted by the Moon from the force exerted by the Earth:
Equilibrant Force = Force (Earth on spaceship) - Force (Moon on spaceship)
Now we can calculate the magnitude of the equilibrant force by taking the absolute value of the result.
Magnitude of Equilibrant Force = |Equilibrant Force|
Plug in the values and calculate:
Magnitude of Equilibrant Force = |(6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.97 × 10^24 kg * 54,100 kg) / (256,000,000 m)^2 - (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (7.35 × 10^22 kg * 54,100 kg) / (384,000,000 m)^2|
By evaluating this expression, we can find the magnitude of the force that the ship's engines would have to supply to be the equilibrant.
To find the magnitude of the force the ship's engines would have to supply to be the equilibrant, we need to first calculate the gravitational force between the Earth and the spaceship at its current position.
The gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 × 10^-11 Nm^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
First, let's calculate the current distance between the spaceship and the Earth. We know that the distance between the Earth and the Moon is 384,000 km, and the spaceship has completed two-thirds of the journey. So, the distance covered by the spaceship is:
Distance = 2/3 * 384,000 km
Distance = 256,000 km
Now, let's convert the distance to meters:
Distance = 256,000 km * 1000 m/km
Distance = 256,000,000 m
Next, we can calculate the gravitational force between the spaceship and the Earth. The mass of the spaceship is given as 54,100 kg, and the mass of the Earth is much greater than the spaceship's mass, so we can neglect the spaceship's mass:
F = (G * m1 * m2) / r^2
F = (6.67430 × 10^-11 Nm^2/kg^2) * (54,100 kg) * (5.972 × 10^24 kg) / (256,000,000 m)^2
Calculating the force:
F ≈ 1.3917 × 10^9 N
Therefore, the magnitude force the spaceship's engines would have to supply to be the equilibrant at this position is approximately 1.3917 × 10^9 Newtons.