Calculating Weight of an Astronaut in Orbit

In summary: So the work done by the rocket is mgh.In summary, a 619 kg satellite in circular orbit 7.84x106 m above the surface of the Earth experiences an acceleration due to gravity of 9.76 m/s2. The weight of a 70.4 kg astronaut inside the satellite can be calculated using the formula Fg = G M(earth) M(astronaut) / r^{2}, where the astronaut's mass is used. To calculate the work done to put the satellite into orbit, the formula W = 1/2 m (satellite) v(orbital)^2 can be used, taking into account the rotation of the Earth.
  • #1
greyradio
12
0
A 619 kg satellite is in circular orbit 7.84x106 m above the surface of the Earth. Find:

a) the acceleration due to gravity created by the Earth at the distance of the satellite.
correct check mark m/s2

b) the weight of a 70.4 kg astronaut inside the satellite.

Equations:
Fg = G M Earth M / r[tex]^{2}[/tex]


Part A is solved. However, part b is simple enough but i seem to be having trouble with it.

My initial attempt was to use the first part of the answer and use Newton's formula F = ma.

However, my next attempt is to use:

Fg = G M(earth) M(astronaut) / r[tex]^{2}[/tex]

However, one thing gives me trouble. Should the astronaut's mass be used or should the astronaut and the satellite. If I'm off track I would appreciate someone's help. Thanks.
 
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  • #2
greyradio said:
However, one thing gives me trouble. Should the astronaut's mass be used or should the astronaut and the satellite.

You don't need the weight of the satellite so you should use the astronaut's mass.

There is also the centrifugal force on the astronaut but you probably don't need it.
 
  • #3
The "weight" of the astronaut is simply the gravitational force on him at that height.
 
  • #4
I have another question relating to this problem

1.the work that was done to put the satellite into this orbit. Assume that it starts at rest on the surface of the earth.

Friction is to be ignored.

I have the orbital speed. Could the work-energy theorem be used?

If so then it would be

W = 1/2 m(satellite) v(orbital)^2 - 1/2 m (0)
W = 1/2 m (satellite) v(orbital)^2

Ive tried using W = F change in distance
where the force is the gravitational force and the distance is the height above the Earth's surface but I got the answer wrong. Perhaps I'm missing something.
 
  • #5
greyradio said:
Perhaps I'm missing something.
You've missed the fact that the Earth is rotating and that the potential energy on the ground and differs from that on orbit.
 
  • #6
Right.

So I would use this formula:

[tex]\Delta[/tex] E = Ef - Ei

So I would get this:

Ef =

1/2 [tex]M_{satellite}[/tex] [tex]V_{orbital speed}[/tex][tex]^{2}[/tex] - G[tex]M_{earth}M_{satellite}[/tex]/R

The radius would be the radius of the Earth plus the height above the surface

Ei = 1/2 [tex]M_{satellite}[/tex] [tex]V_{initial}[/tex][tex]^{2}[/tex] - G[tex]M_{earth}M_{satellite}[/tex]/R

The radius would be the radius of the earth. And the initial velocity would be zero since the satellite is at rest.

Ef - Ei = 1/2 [tex]M_{satellite}[/tex] [tex]V_{orbital speed}[/tex][tex]^{2}[/tex] - G[tex]M_{earth}M_{satellite}[/tex]/R - (- G[tex]M_{earth}M_{satellite}[/tex]/R)
 
Last edited:
  • #7
The satellite is not at rest initially. The Earth is rotating.
 
  • #8
Since the acceleration of the Earth's gravitational force differs at different radial distances from the planet, you need a certain amount of Kinetic Energy to get 7.84x106 m above the surface of the Earth. So the change in kinetic energy of the rocket is equal to negative the change in potential energy...and the work done by the rocket is equal to the change in kinetic energy.
 

1. How is the weight of an astronaut in orbit calculated?

The weight of an astronaut in orbit is calculated using Newton's Second Law of Motion, which states that force is equal to mass times acceleration. In this case, the force is the gravitational force exerted by the Earth on the astronaut, the mass is the astronaut's mass, and the acceleration is due to the Earth's gravity. By rearranging the formula, we can calculate the weight of the astronaut as weight = mass x acceleration due to gravity.

2. Is the weight of an astronaut in orbit the same as their weight on Earth?

No, the weight of an astronaut in orbit is not the same as their weight on Earth. This is because weight is a measure of the gravitational force on an object, and the gravitational force on an object depends on the mass of the object and the distance between the object and the gravitational body. In orbit, the distance between the astronaut and the Earth is much greater than it is on the surface of the Earth, so the gravitational force is much weaker, resulting in a lower weight.

3. How does weightlessness affect the weight of an astronaut in orbit?

Weightlessness does not affect the weight of an astronaut in orbit. Weightlessness is the sensation of not feeling any weight due to the absence of a normal force acting on an object. In orbit, an astronaut is still subject to the gravitational force of the Earth, so they do have weight, but they are in a state of free fall, which creates the sensation of weightlessness.

4. Can the weight of an astronaut in orbit change?

Yes, the weight of an astronaut in orbit can change. This is because the weight of an object is dependent on both the object's mass and the gravitational force acting on it. If an astronaut's mass changes, their weight will also change. Additionally, if the astronaut moves closer or further away from the Earth (such as during a spacewalk), their weight will also change.

5. How is the weight of an astronaut in orbit different from their mass?

The weight of an astronaut in orbit and their mass are two different things. Mass is a measure of the amount of matter in an object and is constant regardless of its location. Weight, on the other hand, is a measure of the gravitational force acting on an object and can vary depending on the object's location. In orbit, an astronaut's mass stays the same, but their weight changes due to the weaker gravitational force.

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