the surface of the earth is a curved surface over large distances and can be tre
ID: 1702986 • Letter: T
Question
the surface of the earth is a curved surface over large distances and can be treated as a sphere spinning on its axis once every 24 hours. The Rochester city is about the latitude of Paris and the kings castle is directly south about the latitude of Paris. radius of the Earth, Re = 6.29 x 106 m).1) Find the perpendicular distance from the axis of the Earth of the Rochester City and the kings castle. Find the angular speed in rad/sec of each of each location on the surface of the Earth. Show a formula for the tangential speed in terms of the angular velocity and calculate each tangential speed.
2) why is two of these values have the same magnitude and two are different.
3) As a traveler moves from the Rochester City directly south to the Kings castle. The traveler will experience a change in tangential velocity and therefore a non zero acceleration. Calculate the magnitude and direction of the average acceleration experienced by the traveler.
Explanation / Answer
1. Paris latitude is approximately 45 degrees, while Romes is approximately 41 degrees. 2. Drawing a circle to represent earth and putting paris and romes in their perspective position around that, make a dot in the middle to represent the earths axis. 3. From the drawing you can now form right triangle to come up with the other two angles. Keep in the mind the radius of the earth is the hypotenuse of these two triangles. 4. Paris perpendicular distance = radius of earth (6.29x10^6m) x Sin(45)= 4.4 x 10^6m 5. Rome perpendicular distance= radius of earth (6.29x10^6m) x Sin(49)= 4.7 x 10^6m 6. Angular speed in radians per second= 2pi/86400 seconds = 7.3 x 10^-5. 86400 seconds is equivalent to 24 hours. 7. Tangential speed (v) = perpendicular distance (r) x angular speed in radian measure Paris (v) = (4.4 x 10^6m) x (7.3 x 10^-5 s) = 321 m/s Rome(v) = (4.7 x 10^6m) x (7.3 x 10^-5 s) = 343 m/s 8. Angular speed is the same because takes time into account, not distance. Tangential speeds are different because distance is a factor in this equation. 9. Using the kinematic equation v2 = vo2 + 2ax, recognize that 321 is final velocity while 343 m/s is initial velocity. Solving for acceleration a = (343)^2- (321)^2/2(4.4 x 10^5) = 1.7 x 10^-2 10. 4.4 x 10^5 from above came from the position equation that says a distance along a circular arc (s) = theta x perpendicular axis (r) in radian measure. If you measure the distance between paris and rome on the circle you would see that it would come out to be 4 degrees. Convert this 4 degrees into radian measure by multiplying it by pi/180= .07 radians. Now, using the equation s = theta x r= (.07) x (6.29 x 10^6)= 4.4 x 10^5m. 11. The direction of acceleration is south. Hope this helps!
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