The earth’s rotation axis, which is tilted 23.5 from the plane of the earth’s orbit, today points to Polaris, the north star. But Polaris has not always been the north star because the earth, like a spinning gyroscope, precesses. That is, a line extending along the earth’s rotation axis traces out a 23.5 cone as the earth precesses with a period of 26,000 years. This occurs because the earth is not a perfect sphere. It has an equatorial bulge, which allows both the moon and the sun to exert a gravitational torque on the earth. Our expression for the precession frequency of a gyroscope can be written Ω=𝜏/ω. Although we derived this equation for a specific situation, it’s a valid result, differing by at most a constant close to 1, for the precession of any rotating object. What is the average gravitational torque on the earth due to the moon and the sun?
Average gravitational torque is 5.34 x 1022 N.m
The tilt of the earth's rotation axis = 23.5o
Period of earth, T=26000 yearsMass of earth, M=5.9 x1024 kg The radius of the earth, R=6400 km The period of the earth on its own axis is t=24 hr
Calculate moment of inertia of earth assuming it is sphere
From the period calculate precision frequency
And natural frequency
we can calculate torque as
Torque = precision frequency x moment of inertia x natural frequency
Blocks of mass m1 and m2 are connected by a massless string that passes over the pulley in Figure P12.65. The pulley turns on frictionless bearings. Mass m1 slides on a horizontal, frictionless surface. Mass m2 is released while the blocks are at rest. a. Assume the pulley is massless. Find the acceleration of m1 and the tension in the string. This is a Chapter 7 review problem.b. Suppose the pulley has mass mp and radius R. Find the acceleration of m1 and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set mp = 0.
Your task in a science contest is to stack four identical uniform bricks, each of length L, so that the top brick is as far to the right as possible without the stack falling over. Is it possible, as Figure P12.61 shows, to stack the bricks such that no part of the top brick is over the table? Answer this question by determining the maximum possible value of d.
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