Question

The earth (of radius of 6371km) rotates on itself at a rate of one revolution every 24 hours and is titled by an angle of 23.5°. Let's assume that the earth is a perfect sphere, which allows us to consider a uniform circular motion 23.5 R equator Figure 1: Earth rotation Everybody on earth share the same time of rotation, but based on their location on the earth, they wil have different velocities. A person located in Hawaii, at an angle of 21.3° above the equator will have different linear velocity than a person located at the equator. The difference in velocities comes from the fact that the radius around which they rotate is different, as show in Figure (1). Assuming that the earth is spherical, that there is no sinificant variation in the earth's rotation over a complete cycle, and using the fundamental definition of velocity: dr dt 3) Calculate the centripetal acceleration of a person on the equator (located at point E in Figure 1)) and of a person in Hawaii (located at point H in Figure (1)) Express your answers in SI units (m/s2). Please note that the above acceleration is strictly due to the rotation of the earth and has a direction towards perpendicular to the earth's axis of rotation

Answer 1

3)

Centripetal acceleration in a circular motion is given by

$a_c=\omega^2r$

as we can see in the motion for both persons the angular velocity will remain same however the radius of circular motion is different.

total angle 2*π-=7.27 * 10-5rad/sec time taken 3600 24

radius for the person on equator is same as that of radius of earth=6371000m

while for the person is Hawai it is different. It can be calculated using simple trigonometry.

Look at the angle between earth radius line and the line subtended to hawai surface. It is 21.3^o.

Radius for hawai

Rr 21. 21.3

Rh = R cos 21.30

So

for equator

= w2R = 6371000 * (7.27 * 10-5)2-0.0337m/s

for hawai person

ac = a,2R cos 21.3° = 6371000 * (7.27 * 10-5)2 * 0.93 = 0. 2 0313m/s