OpenStax College Physics for AP® Courses, Chapter 10, Problem 16 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. The archenemy of Superman named Zorch is threatening to slow down the rotation of the Earth so that the Earth's period will be 28.0 hours instead of 24.0 hours and Zorch will exert a force that's tangential to the Earth or in other words, perpendicular to the radius and at the equator and so the lever arm then is going to be the radius of the Earth at the equator and the rotation of the Earth is in this direction looking down from the North Pole you know, this would be New York city here and the Sun might be here and New York city will see the Sun first and then it will rotate out of the way and then Los Angeles will eventually get the Sun later so the Sun rises in New York before Los Angeles for example. Okay! So there's a given force here: the force that a Titan V rocket can exert or a Saturn V rocket I should say and how long will it take for this torque to result in a change in angular speed such that the period goes from 24 hours to 28 hours. So final angular velocity is the initial angular velocity plus angular acceleration multiplied by time and we can solve this for t by subtracting ω naught from both sides and then dividing both sides by α so we get t is final angular velocity minus initial divided by angular acceleration so we need to figure out each of these terms. So the final angular velocity is the number of radians per second and so that's one revolution of the Earth, or rotation you could say, times 2πrad for every revolution and that happens in 1.008 times 10 to the 5 seconds because that's 28 hours converted into seconds by multiplying by 3600 seconds per hour. So that's 6.2333 times 10 to the minus 5 radians per second— it's the final angular velocity of the Earth— the initial angular velocity is 2πrad divided by 8.64 times 10 to the 4 seconds—that's 24 hours expressed in seconds— and that's 7.2722 times 10 to the minus 5 radians per second. So we have substitutions now for each of these terms and then we need to work on this angular acceleration term now or factor I should say. Okay! So we have net torque is the force times lever arm and these are perpendicular in this picture so we just multiply them together to get that torque and then we also have an expression for net torque as angular acceleration multiplied by moment of inertia and so that means we can equate these two things because they are both equal to the net torque and we do that here. I mean divide both sides by I and we get that the angular acceleration then is the torque divided by moment of inertia. So to substitute for I, we need to consult our list of formulas and we are going to be using this one because we have a solid sphere— the Earth is solid... I mean the inside is somewhat liquidy but that's good enough... it's certainly not hollow, it's not a shell— so it's a solid sphere rotating about an axis which is a diameter and so the formula is going to be 2 times its mass times its radius squared divided by 5. So we are going to divide by this fraction but instead of dividing by fractions, I like to multiply by the reciprocal of the fraction so I am multiplying Fr by 5 over 2mr squared, this r cancels with one of these r's and we are left with 5F over 2mr. So the angular acceleration then is 5 times 4.00 times 10 to the 7 Newtons divided by 2 times the mass of the Earth times the Earth radius at the equator and this gives 2.6254 times 10 to the minus 24 radians per second squared. So now we finally have substitutions for each of these things. So we substitute the final angular velocity minus the initial angular velocity and then substitute the angular acceleration and I put a negative sign on there because this, you know, is in the opposite direction to the initially positive angular velocity and so being in the opposite direction to this positive thing that makes this the negative thing. This works out to 3.96 times 10 to the 18 seconds which we convert into years in order to understand this time better so we multiply by 1 hour for every 3600 seconds and then by 1 day for every 24 hours and then by 1 year for every 365 days and we end up with 1.25 times 10 to the 11 years. This is longer than the age of the universe which has existed for only about 14 billion years, which is 1.4 times 10 to the 10 years so this is one order of magnitude longer than the age of the universe and so Superman can take care of other villains for a long, long time before he needs to worry about Zorch.