Engineers are trying to create artificial "gravity" in a ringshaped space station by spinning it like a centrifuge. The ring is 170 m in radius. How quickly must the space station turn in order to give the astronauts inside it apparent weights equal to their real weights at the earth’s surface?

To determine the necessary angular velocity at which the space station must turn to create artificial "gravity" equal to the astronauts' real weights at the Earth's surface, we can utilize the concept of centripetal force.

First, let's establish the formula for centripetal force:

F = m * ω² * r

Where:
F = Centripetal force
m = Mass of the object
ω = Angular velocity
r = Radius of rotation

In this case, we want the centripetal force to equal the weight of the astronauts when they are standing on the Earth's surface. Therefore, we can set the centripetal force equal to the gravitational force acting on them:

F = mg

Where:
m = Mass of the astronaut
g = Acceleration due to gravity

Now, let's substitute the equation for centripetal force in terms of gravitational force:

mg = m * ω² * r

The mass (m) of the astronaut cancels out, leaving:

g = ω² * r

Finally, solving for the angular velocity (ω) gives us:

ω = √(g / r)

Now we can plug in the given values:
radius (r) = 170 m
acceleration due to gravity (g) = 9.8 m/s²

ω = √(9.8 m/s² / 170 m)

Calculating this, we find:

ω ≈ 0.487 rad/s

Therefore, the space station must turn at an angular velocity of approximately 0.487 radians per second to create artificial "gravity" that gives the astronauts inside apparent weights equal to their real weights at the Earth's surface.