Consider a space station that is rotating to create artificial gravity. Calculate the minimum speed required for an object to break free from the surface of the rotating space station.
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To calculate the minimum speed required for an object to break free from the surface of a rotating space station, we need to consider the centripetal force acting on the object and compare it to the gravitational force between the object and the space station.
The centripetal force acting on an object rotating in a circle is given by:
Fc = m * (v^2 / r)
Where:
- Fc is the centripetal force
- m is the mass of the object
- v is the linear speed of the object
- r is the distance from the center of rotation to the object
The gravitational force between the object and the space station is given by:
Fg = m * g
Where:
- Fg is the gravitational force
- m is the mass of the object
- g is the acceleration due to gravity
For the object to break free from the surface, the centripetal force needs to be equal to or greater than the gravitational force. So we can set up the following inequality:
Fc ≥ Fg
m * (v^2 / r) ≥ m * g
Canceling out the mass (m) on both sides, we get:
v^2 / r ≥ g
Multiply both sides by r:
v^2 ≥ g * r
Finally, take the square root of both sides to solve for v:
v ≥ √(g * r)
Therefore, the minimum speed required for an object to break free from the surface of the rotating space station is equal to or greater than √(g * r), where g is the acceleration due to gravity and r is the distance from the center of rotation to the object.
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