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Chapter 5: Problem 64

You stand on a spring scale at the north pole and again at the equator. Which scale reading will be lower, and by what percentage will it be lower than the higher reading? Assume \(g\) has the same value at pole and equator.

Short Answer

Expert verified
The reading on the spring scale at the equator will be lower by 50 percent compared to the reading at the north pole.

Step by step solution

01

Understand the difference in forces at the north pole and the equator

On the earth's surface at the pole, only the gravitational force is present, pulling downwards. However, on the surface at the equator, there are two forces acting on a body. One is the gravitational force pulling downwards, and the other is the centrifugal force from the rotation of the Earth, pulling outwards or upwards.
02

Define and find the formula for forces at pole and equator

At the north pole, where the centrifugal force is negligible, the force measured by the scales is purely the gravitational force, \(F_1 = mg\). At the equator, it is the difference between the gravitational force and the centrifugal force, \(F_2 = mg - m\omega^2R\), where \(m\) is the mass of the object, \(\omega\) is the angular velocity and \(R\) is Earth's radius. Because the gravitational and centrifugal forces are equal at the equator, we know that \(g = \omega^2R\), and we can substitute this into the equation for \(F_2\), to get \(F_2 = g(m - \frac{m}{2}) = \frac{mg}{2}\).
03

Calculate the percentage by which the equatorial reading is lower

The percentage by which the equatorial scale reading \(F_2\) is lower than the polar scale reading \(F_1\) is given by the formula: \((F_1 - F_2) / F_1 * 100 = (\frac{mg}{2} / mg) * 100 = 50% \)

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