A kg ball on the end of a lightweight rod is located at where the axis is vertical. The other end of the rod is attached to a pivot at . What is the torque about the pivot? Write your answer using unit vectors.
Hence, the torque applied on the mass is
The rotating equivalent of force is torque. It is provided by.
The force is , and the distance between the pivot and the site of application of force is .
The right-hand thumb rule determines the direction of torque, which is a vector quantity.
For the application of force, see the diagram below:
The position vector of the mass is:
Substitute for and for
On the mass, the force vector is:
Here, and are the mass and acceleration of the ball, respectively.
The torque acting on mass is
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?
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