Question 1133892: Question: The arc from Point A to the North Pole of a planet subtends a central angle of 45 degrees. The radius of the planet is 4450 mi. Any point on the surface of the planet (except at the poles) makes one revolution (2pi radians) about the axis of the planet in 26 hours. What are the angular and linear velocities for Point A with respect to its rotation around the axis of the planet?
I already know that the angular velocity is pi / 13. I just don't know how to get the other radius for linear velocity.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! You have the correct angular speed.
angular speed = (amount of radians in a full revolution)/(amount of time for one full revolution)
angular speed = (2pi radians)/(26 hours)
angular speed = (pi/13) radians per hour
Nice work on getting that answer.
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Unfortunately determining the linear speed takes a bit more work than what was done above. Let's start off drawing out the picture
Point A is the location mentioned in the instructions (some city perhaps).
Point B is the north pole
Point C is the center of the planet
Line BC is the axis of rotation for the planet.
Segment AE is perpendicular to BC. The two lines cross to form point D.
Central angle ACB is 45 degrees. This is given in the instructions.
The ultimate goal is to find the length of segment AD which is why I highlighted it in red. This is the radius of the circular path point A traces out as the planet does one full revolution, similar to what is shown below
(image credit: WikiMedia.org)
This is known as a circular cross section of a sphere. The goal we want is to find the value of lowercase r, which is the radius of that red shaded circle (contrast this with the uppercase R: the radius of the sphere)
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Before we can find AD, we must find the length of AB. Use the law of cosines. Focus on triangle ABC. Ignore point D and point E for now.
a = 4450
b = 4450
c = unknown
angle C = 45 degrees
c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = 4450^2 + 4450^2 - 2*4450*4450*cos(45)
c^2 = 11,600,035.9311068
c = sqrt(11,600,035.9311068)
c = 3405.8825480493
So AB is roughly 3405.8825480493 miles long.
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Use this to find the measure of angle ABC. Use the law of sines. Focus only on triangle ABC. This allows us to write uppercase B in place of saying "angle ABC".
sin(B)/b = sin(C)/c
sin(B)/4450 = sin(45)/3405.8825480493
sin(B)/4450 = 0.0002076133781
sin(B) = 4450*0.0002076133781
sin(B) = 0.923879532545
B = arcsin(0.923879532545)
B = 67.5000000050477
Angle ABC is roughly 67.5000000050477 degrees. In this case, angle ABC is the same as angle ABD.
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Finally we can determine the length of AD using the sine trig ratio. This ratio shown below only works for right triangles, which triangle ABD happens to be. Focus only on this triangle (ignore points C and E)
sin(angle) = opposite/hypotenuse
sin(B) = AD/AB
sin(67.5000000050477) = AD/3405.8825480493
0.923879532545 = AD/3405.8825480493
3405.8825480493*0.923879532545 = AD
AD = 3146.62517639497
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We have the radius of the circular cross section we want. This radius is roughly 3146.62517639497 miles. The circumference of this circle cross section is...
Circumference = 2*pi*r
Circumference = 2*3.14*3146.62517639497
Circumference = 19760.8061077604
Point A traces out this approximate distance as the planet spins one full revolution. It does this in 26 hours, so the linear speed is
linear speed = distance/time
linear speed = (19,760.8061077604 miles)/(26 hours)
linear speed = 760.03100414463 miles per hour
This answer is approximate. Round it however you need to.
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