It has been suggested that rotating cylinders about 10 mi long and 5.0 mi in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration on Earth

Centripetal acceleration is given by:

a = v²/r

a = centripetal acceleration, v = speed of cylinder at surface, r = radius of cylinder

This equation relates the speed of the cylinder at its surface to its angular velocity:

v = rω

v = speed at surface, r = radius, ω = angular velocity

Make a substitution:

a = (rω) ²/r

a = rω²

Given values:

a = 9.81m/s², r = (5.0mi) / 2 = 2.5mi = 4023m

Plug in and solve for ω:

9.81 = 4023ω²

ω = 0.049rad/s