It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolution

Question

It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolution (mean sidereal day). What is the angular speed of the earth?

Assume the earth is spherical. Relative to someone on the rotation axis, what is the linear speed of an object on the surface if the radius vector from the center of the earth to the object makes an angle of 37.0° with the axis of rotation. The radius of the earth is 6.37×103 km.

What is the acceleration of the object on the surface of the earth in the previous problem?

Please show all work, an answer in sig-figures

Thank you for all your help.

Explanation / Answer

the angular speed, ? = angle / time for complete one revolution, ? = 2p or 360 degree and time = 23 h + 56 min + 4 s = 86164s ? ? = 360/86164 degree/s = 0.004178 degree/s = 7.29×10^5 radius/s --------------------------------- The radius of the earth, r = 6.37×10^3 km But now we are not on the equator but 37.0° south from the pole. Therefore we need a new "radius" r' for our location. r' = r*Sin[37.0°] = 3.834×10^3 km the Linear speed, v = r ? here ? is the angular speed in rad/s. ? v = 7.29×10^5*3.834×10^3 = 0.27958 km/s = 279.58 m/s -------------------------------------- Without considering the gravitational force, the acceleration of the object , a = v^2/r = (279.58) ^2 / 3.834×10^6 = 0.02039 m/s^2

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