# Find period of astronaut when Normal force of ship = normal force on earth

• kariibex
In summary, to show that the period of rotation of a spaceship needed to make an astronaut feel the same normal force as on Earth is given by T = √[(4pi^2)(Rs)(Re^2)/GMe], you can substitute the given equations for normal force and velocity into the formula for period of rotation, T = 2∏r/v. This results in T = √[(4pi^2)(Rs)(Re^2)/GMe], where Rs is the radius of the spaceship and Re is the radius of Earth.
kariibex

## Homework Statement

If the gravitational acceleration at Earth's surface is given by: g = GMe/Re^2, show that the period of rotation of a spaceship needed to make an astronaut feel the same normal force on the ship that they would feel on Earth, when it is given by T = √[(4pi^2)(Rs)(Re^2)/GMe], where Rs is the radius of ship and Re is radius of earth.

## Homework Equations

Fc = ma = m(v^2/r)
Fg = gMem/r^2
v = 2∏r/ T

## The Attempt at a Solution

Normal force on Earth = normal force on ship
N = mg = m(GMe/Re^2)
Normal force ship on astronaut = mac = m(v^2/r)

Now I'm not sure what to substitute into the Period formula and how to approach this?

Rewrite v = 2∏r/ T into the form T = ... You have an expression for v, and r = Rs. Just substitute in.

## 1. What is the significance of finding the period of an astronaut when the normal force of the ship is equal to the normal force on earth?

The period of an astronaut refers to the time it takes for one complete orbit around the earth. When the normal force of the ship is equal to the normal force on earth, it means that the gravitational force experienced by the astronaut is the same as that experienced on the surface of the earth. This can be used to calculate the period of the astronaut's orbit.

## 2. How is the period of an astronaut related to the normal force of the ship and the normal force on earth?

The period of an astronaut is directly proportional to both the normal force of the ship and the normal force on earth. This means that as the normal force of the ship or the normal force on earth increases, the period of the astronaut's orbit also increases.

## 3. What factors affect the normal force of the ship and the normal force on earth?

The normal force of the ship and the normal force on earth are affected by the mass of the astronaut, the mass of the ship, and the distance between the astronaut and the center of the earth. The gravitational force experienced by the astronaut is also affected by the mass and radius of the earth.

## 4. How is the period of an astronaut affected by changes in the normal force of the ship or the normal force on earth?

As mentioned earlier, the period of an astronaut's orbit is directly proportional to the normal force of the ship and the normal force on earth. This means that any changes in these forces will result in a change in the period of the astronaut's orbit. For example, if the normal force on earth is doubled, the period of the astronaut's orbit will also double.

## 5. Can the period of an astronaut's orbit be calculated using the formula for gravitational force?

Yes, the period of an astronaut's orbit can be calculated using the formula for gravitational force, which is F = (G * m1 * m2) / r^2. In this formula, G is the gravitational constant, m1 and m2 are the masses of the astronaut and the earth respectively, and r is the distance between the astronaut and the center of the earth. By rearranging this formula, the period of the astronaut's orbit can be calculated as T = 2π * √(r^3 / G * m2).

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