(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6.4×106 m6.4 \times 10^{6}\textrm{ m} at its equator, what is the linear velocity at Earth's surface?
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Final Answer
  1. 86500 s86500 \textrm{ s}
  2. 7.27×105 rad/s7.27 \times 10^{-5}\textrm{ rad/s}
  3. 470 m/s470 \textrm{ m/s}

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OpenStax College Physics, Chapter 6, Problem 4 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. The period of Earth's rotation is the number of seconds it takes for 1 day. So we multiply by 24 hours per day leaving us with units of hours times by 60 minutes per hour leaving us with units of minutes and then convert that into seconds by multiplying by 60 seconds per minute. So we are left with seconds as a unit and so we have 1 times 24 times 60 times 60 which is 86400 seconds per day. The angular speed of the Earth is the number of radians through which it rotates which is 2π radians— that's 1 full day of rotation— divided by the time it takes to do that which is 86400 seconds. So the angular speed then is 7.27 times 10 to the minus 5 radians per second. The linear speed on a point at the equator on the surface of the Earth is going to be the radius of the Earth multiplied by this angular speed. So that's 6.4 times 10 to the 6 meters—radius— times 2π radians over 86400 seconds which is 470 meters per second.


For part C. Quick question, why would we not use 360 in the place of change of theta instead of 2 (π)? Thank you in advance.

Hi phillipmoreno,
Thank you for the question. Degrees are not considered mks units. mks refers to meters, kilograms, seconds, which are the units needed for formulas. Another way to look at it is that, since radians are the ratio of the distance traveled along the edge of a circle divided by the circle's radius, radians tell you how many radii have been traveled (the name radians could instead be expressed "number of radii". 2π2\pi radians is saying 6.28 radii in other words). When we divide the "number of radii" travelled by time, we get units of "radii per second", which then gets multiplied by "meters per radius", arriving at "meters per second". I'm putting strange units in quotes there, but I'm trying to help conceptualize the meaning of radians, and why they're helpful in formulas.
All the best,