To calculate the escape velocity required for an object to break free from the surface of a rotating space station, we can use the concept of rotational kinetic energy and gravitational potential energy.

The rotational kinetic energy of the space station can be given by:

K_rot = (1/2) I ω^2

Where I is the moment of inertia of the rotating space station and ω is the angular velocity. Since we know the radius of the space station and it is rotating uniformly, we have

ω = v / r

Where v is the linear velocity of the space station and r is the radius.

The moment of inertia of a hollow sphere is given by:

I = (2/3) m r^2

Where m is the mass of the space station.

Now, the gravitational potential energy of an object on the surface of the space station is:

U = -G m M / R

Where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.

Now, for an object to escape the space station, its kinetic energy (rotational and linear) must be greater than or equal to the absolute value of its potential energy. So, we have:

K_rot + (1/2) m v^2 ≥ |U|

Substituting the above equations, we get:

(1/2) I ω^2 + (1/2) m v^2 ≥ G m M / R

Rearranging the equation, we get:

[(1/2) I ω^2] / m + (1/2) v^2 ≥ G M / R

Substituting ω = v / r and I = (2/3) m r^2:

[(1/2) (2/3) m r^2 (v / r)^2] / m + (1/2) v^2 ≥ G M / R

(1/3) v^2 + (1/2) v^2 ≥ G M / R

(5/6) v^2 ≥ G M / R

v^2 ≥ (6/5) (G M / R)

Taking the square root of both sides, we get:

v ≥ sqrt(6/5) sqrt(G M / R)

Now, we can substitute the given values:

G ≈ 6.67430 × 10^-11 m^3 kg^−1 s^−2 (gravitational constant)
M ≈ 5.972 × 10^24 kg (mass of the Earth)
R = 6.371 × 10^6 m (radius of the Earth)

Applying these values, we can calculate the escape velocity:

v ≥ sqrt(6/5) sqrt((6.67430 × 10^-11) × (5.972 × 10^24) / 6.371 × 10^6)

v ≥ sqrt(6/5) sqrt(6.67430 × 10^-5)

v ≥ sqrt(6/5) × 0.0081598

v ≥ 0.008994 m/s

Therefore, the escape velocity required for an object to break free from the surface of the rotating space station, where the radius decreases from 200 meters to 100 meters, is approximately 0.008994 m/s.

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