Mike Dworetsky wrote:
Yousuf Khan wrote:
On 20/06/2013 10:28 PM, wrote:
How fast would the earth have to spin, so that stuff
started flying off?


Given the Earth's radius is 6367.5 km, a 1 kg mass, being held down
by 1 N of gravitational force would need to balance out that 1 N of
gravity with an equal and opposite 1 N centripetal force, at a
minimum. Therefore the Earth would need to rotate at an incredible
speed of rotation speed of 2.523 km/s = 5645 mph = 9084 km/h at the
equator!
The Earth's actual current rotational speed is only 0.4651013 km/s =
1040.4 mph = 1674.365 km/h. This results in a miniscule centripetal
force of only 0.03397 N, against the Earth's gravitational force of 1
N on a 1 kg mass.

Yousuf Khan

You are forgetting the Earth's gravitational acceleration in the
equation! Earth's gravitational force on a 1 kg mass is 9.8 N. g =
9.8 m/s^2.
Hence for balance, mg = m v^2/r or, circular velocity v for balance of
centripetal acceleration and gravitational attraction is v = sqrt (rg) =
7900 m/s or 7.9 km/s. For a circumference of 40000 km, the
rotation period would need to be 1.406 hours (5063 sec).

The reduction in weight at the equator vs the poles for today's 23h56m
rotation period (neglecting obliquity of the Earth) is about 0.0034
or 0.34 %.

Thinking about it for a moment, the required speed would be the same as that
needed for reaching orbital velocity at altitude zero. If there were no
atmosphere that would work, so stuff would start "flying off", but only at
the equator.

--
Mike Dworetsky

(Remove pants sp*mbl*ck to reply)