A satellite has a mass of 5850 kg and is in a circular orbit 4.50 x 10^5 m above the surface of a planet. The period of the orbit is two hours. The radius of the planet is 4.20 x 10^6 m. What is the true weight of the satellite when it is at rest on the planet's surface?
Ah, the true weight of the satellite when it's at rest on the planet's surface? Well, I'm assuming the satellite doesn't have a penchant for lying about its weight, so we can simply calculate it!
First, let's find the gravitational force between the planet and the satellite in its orbital position. We can use the formula:
F = (G * m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between the centers of the masses.
Now, since the satellite is in a circular orbit, the gravitational force between the satellite and the planet is providing the necessary centripetal force to keep the satellite in orbit. So, we have:
F = (m * v^2) / r
Where m is the mass of the satellite, v is the orbital velocity, and r is the distance between the satellite and the center of the planet.
Solving for v:
v = (2 * pi * r) / T
Where pi is our beloved mathematical constant and T is the period of the orbit.
Next, let's find the value of v using the given values. We can plug in the numbers and calculate the orbital velocity.
v = (2 * 3.14 * 4.50 x 10^5) / (2 * 3600)
After some calculating, we find the orbital velocity to be approximately 1962.3 m/s.
Now, let's substitute the values of m, v, and r into our formula for F:
F = (m * v^2) / r
F = (5850 * (1962.3)^2) / (4.50 x 10^5)
Again, after some calculations, the gravitational force F comes out to be approximately 203,969.7 N.
But hold your laughter, we're not done yet!
You see, the true weight of an object is just the force it experiences due to gravity. So, the true weight of the satellite when it's at rest on the planet's surface is equal to the gravitational force it experiences at that location.
However, since the satellite is already at rest on the planet's surface, we know that the gravitational force on the satellite and its weight are equal. Therefore, the true weight of the satellite when it's at rest on the planet's surface is approximately 203,969.7 N.
Voila!
To find the true weight of the satellite when it is at rest on the planet's surface, we need to find its weight in the given circular orbit and then adjust it for the difference in gravitational pull between the surface and the orbit.
Step 1: Find the gravitational force acting on the satellite in its circular orbit.
The gravitational force can be calculated using the formula:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Let's assume the mass of the planet is much larger than the satellite's mass (which is typically true for satellites orbiting planets), so we can treat the planet as fixed.
The distance between the center of the planet and the satellite is given as 4.50 x 10^5 m + 4.20 x 10^6 m = 4.65 x 10^6 m (height above the surface + radius of the planet).
Plugging in the values into the formula, we can calculate the gravitational force acting on the satellite:
F = (6.674 × 10^-11 N m^2/kg^2) * (5850 kg) * (mass of the planet) / (4.65 x 10^6 m)^2
Step 2: Find the mass of the planet.
The mass of the planet is not provided in the question. We need this information to calculate the gravitational force. Without this information, we cannot find the true weight of the satellite on the planet's surface.
Assuming we know the mass of the planet, we can proceed to calculate the gravitational force.
Step 3: Adjust the gravitational force for the difference in gravitational pull.
When the satellite is at rest on the planet's surface, the gravitational acceleration acting on it is equal to the acceleration due to gravity on the planet. We can find this by dividing the gravitational force on the satellite by its mass.
Weight = (G * m1 * m2) / r^2 / m1
Step 4: Calculate the true weight of the satellite on the planet's surface.
Finally, we can calculate the true weight of the satellite on the planet's surface by substituting the values into the equation obtained in step 3.
To find the true weight of the satellite when it is at rest on the planet's surface, we need to calculate the gravitational force acting on the satellite.
The formula to calculate the gravitational force is given by Newton's law of universal gravitation:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.674 x 10^-11 N*m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
In this case, the satellite is at rest on the planet's surface, so its acceleration is zero, and the weight of the satellite is equal to the gravitational force acting on it.
Now, let's break down the steps to calculate the true weight of the satellite:
1. Determine the mass of the planet:
The mass of the planet is not given directly in the question. However, we can use the radius of the planet to calculate it using the formula for the volume of a sphere:
V = (4/3) * π * r^3
From the volume, we can calculate the mass using the density of the planet (which is not given). So, we will assume a density here.
2. Calculate the radius of the satellite's orbit from the surface of the planet:
The radius of the orbit around the planet is given as 4.50 x 10^5 m above the surface of the planet. To find the total radius, we add the radius of the planet to the given value.
Total radius = radius of the planet + height of the satellite's orbit above the planet's surface
3. Calculate the speed of the satellite:
The speed of the satellite can be calculated using the formula for the circumference of a circle:
C = 2 * π * r
Time period = 2 hours = 2 * 60 minutes = 2 * 60 * 60 seconds
Speed = Circumference / Time period
4. Determine the gravitational force acting on the satellite:
Using Newton's law of universal gravitation, we can calculate the gravitational force acting on the satellite.
5. Convert the gravitational force into the true weight of the satellite:
The true weight of the satellite is the force acting on it due to gravity. This force can be calculated by multiplying the mass of the satellite by the acceleration due to gravity on the planet's surface.
Now, let's perform these calculations step by step:
Step 1: Determine the mass of the planet
- Calculate the volume of the planet using the formula: V = (4/3) * π * r^3
- Calculate the mass of the planet using the density of the planet (assume a value if not given)
Step 2: Calculate the radius of the satellite's orbit from the surface of the planet
- Add the radius of the planet to the given height of the satellite's orbit above the planet's surface
Step 3: Calculate the speed of the satellite
- Calculate the circumference of the satellite's orbit using the formula: C = 2 * π * r
- Convert the time period from hours to seconds
- Calculate the speed using: Speed = Circumference / Time period
Step 4: Determine the gravitational force acting on the satellite
- Use Newton's law of universal gravitation formula: F = (G * m1 * m2) / r^2
- Substitute the masses of the satellite and the planet, and the total radius of the orbit
Step 5: Convert the gravitational force into the true weight of the satellite
- The true weight of the satellite is the force acting on it due to gravity, which is equal to the gravitational force calculated in Step 4
By following these steps, you will be able to find the true weight of the satellite when it is at rest on the planet's surface.