A satellite has a mass of 5750 kg and is in a circular orbit 3.6 * 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 * 10^6 m. What is the true weight of the satellite when it is at rest on the planet's surface?
Ok for this i know we have to use F = ma, and we also use the formula Apparent weight = mg + ma. And that F = (G m1m2)/r^2 plus ac which is v^2/r...now how do i get this started..im actually pretty confused what to do
Yep, you are confused. At orbit, the satellite is weightless, that leads to the conclusion the force of gravity equals the centripetal force.
mv^2/r= GMp m/r^2
note that v= 2PI r/Period.
Put that in for v, divide out the m, divide out a lot of r, and solve for G*Mp.
Now, weight=GMp 5720/(4.2E6)^2
hold on how do i find the mass of the planet?
I suggested you solve for G*Mp
mv^2/r= GMp m/r^2
(2pi*r)^2/r= GMp /r^2
GMp= 4pi^2 r^3
If you want Mp, divide each side by G.
You are given r, the radius to the orbit.
To find the true weight of the satellite when it is at rest on the planet's surface, we need to calculate both the gravitational force and the centripetal force acting on the satellite while in orbit.
Let's break down the problem into steps:
Step 1: Calculate the mass of the planet.
Since we are given the radius of the planet, we can use the formula for the volume of a sphere to find the mass. The volume of a sphere is given by V = (4/3)πr^3, where r is the radius. Knowing the density of the planet, let's assume it is uniform, we can express the mass of the planet as M = ρV, where ρ is the density.
Step 2: Calculate the gravitational force between the planet and the satellite.
Using Newton's law of universal gravitation, the formula for the gravitational force between two masses is F = G(m1m2)/r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Step 3: Calculate the centripetal force on the satellite.
The centripetal force acting on an object moving in a circular path can be calculated using the formula F = mv^2/r, where m is the mass of the object, v is its velocity, and r is the radius of the circular path.
Step 4: Calculate the true weight of the satellite.
The true weight of the satellite is the difference between the gravitational force and the centripetal force, as the centripetal force counteracts some of the gravitational force in order to maintain the satellite in its circular orbit.
Finally, let's go through each step and solve the problem:
Step 1: Calculate the mass of the planet.
We are not given the density of the planet, so we cannot calculate its mass. However, since we only need the true weight of the satellite when it is at rest on the planet's surface, we can assume that the mass of the planet is not needed for this calculation.
Step 2: Calculate the gravitational force between the planet and the satellite.
Using the formula F = G(m1m2)/r^2, where G is the gravitational constant (6.67 × 10^-11 N(m/kg)^2), m1 is the mass of the planet, m2 is the mass of the satellite, and r is the distance between their centers, we can calculate the gravitational force.
F_gravity = G(m1m2)/r^2
Step 3: Calculate the centripetal force on the satellite.
Using the formula F = mv^2/r, where m is the mass of the satellite, v is its velocity, and r is the radius of the circular orbit, we can calculate the centripetal force.
F_centripetal = mv^2/r
Step 4: Calculate the true weight of the satellite.
The true weight of the satellite is the difference between the gravitational force and the centripetal force.
True weight = F_gravity - F_centripetal
Using these steps, you can plug in the given values and solve for the true weight of the satellite when it is at rest on the planet's surface.