A). It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolut
ID: 1420467 • Letter: A
Question
A). 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?
B). 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 49.0° with the axis of rotation. The radius of the earth is 6.37×103km.
C). What is the acceleration of the object on the surface of the earth in the previous problem?
Explanation / Answer
The red arrow is the net acceleration, ignore this. Split the centripetal acceleration into its vectors (1 in line with g (call it x) and 1 perpindicular to it (call it y) )
R = surface radius = 6.37 e6 meters
m = mass of earth = 5.972 e24 kg
G = constant = 6.67384 e-11
g = gravitational acceleration of object = ( G * m ) / R ² = 9.82238 m/s/s
23 h 56 m 4 s = 86,164 seconds
Angular rotation (w) = ( 2 * pi ) / 86,164 = 7.29212 e-5 rad/sec (ANSWER A)
r = object radius from earth axis = cosine 41 ° * R = 4.8075 e6 meters
v = tangential velocity of object = w * r = 350.56 m/s (ANSWER B))
a = centripetal acceleration of object = v ² / r = 0.02556 m/s/s
The vector (x) of the centripetal acceleration that acts directly against the gravitational acceleration :
= cosine 41 ° * 0.02556 = 0.01929 m/s2
The net acceleration acting toward the earths mass center = 9.82238 - 0.01929 = 9.803 m/s/s
( ANSWER C) )
The sideways acceleration vector (y) = sine 41 ° * 0.02556 = 0.01676 m/s/s