What Is the Weight of a Satellite Orbiting Mars?

[GrkFizzix]

Hi Guys I am in year 12 and have my exam shortly would just like to ask a question to do with gravity.

A small satellite orbits Mars. It has a kinetic energy of 3.0x10^10 J, and is at a constant distance of 8.0x10^7 m from the center of Mars. What is the weight of the satellite at this height?

 

[ehild]

How is weight defined?

ehild

 

[GrkFizzix]

it gives no further explanation but I am guessing mg?

 

[ehild]

No, mg is at the surface of Earth. I asked what you have learned about weight, as there are two different definitions and one says that the orbiting bodies are weightless. Here, I think weight is meant as the gravitational force of Mars on the satellite.

To get the force of gravity at distance R from Mars, you can use the equations for the kinetic energy related to the centripetal force along a circular orbit. If m is the mass of the satellite and M is that of Mars, and G is the gravitational constant, the gravitational force is GmM/R^2. The kinetic energy is KE= .5 mv^2.
For a circular orbit, the centripetal force = the force of gravity, and this is the weight of the satellite.

The centripetal force is mv^2/R,

mv^2/R = GmM/R^2.

Go ahead.


ehild

 

[rdl]

I would approach this question by using the formula for gravitational potential energy: U = mgh, where m is the mass of the satellite, g is the acceleration due to gravity on Mars, and h is the height of the satellite from the surface of Mars. Since the satellite is in a constant orbit, we can assume that its potential energy is equal to its kinetic energy. Therefore, we can rearrange the formula to solve for the mass of the satellite: m = U/gh.

Next, we can use Newton's law of universal gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This can be expressed as F = GmM/d^2, where G is the universal gravitational constant, M is the mass of Mars, and d is the distance between the center of Mars and the satellite.

By combining these two equations, we can calculate the weight of the satellite at this height by multiplying the mass of the satellite by the acceleration due to gravity on Mars. This would give us the force of gravity acting on the satellite, which we can then convert to weight using the formula W = mg.

It is important to note that the weight of the satellite may vary slightly depending on its exact location in its orbit, as the distance from the center of Mars may change slightly. However, this calculation should give us a good estimate of the weight of the satellite at a constant distance of 8.0x10^7 m from the center of Mars. I hope this helps and good luck on your exam!