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Chapter 12: Problem 25

A plumb bob located at latitude \(55.0^{\circ} \mathrm{N}\) hangs motionlessly with respect to the ground beneath it. \(A\) straight line from the string supporting the bob does not go exactly through the Earth's center. Does this line intersect the Earth's axis of rotation south or north of the Earth's center?

Short Answer

Expert verified
Answer: South of the Earth's center.

Step by step solution

01

Clarify the effect of gravity and centrifugal force on the plumb bob

The hanging plumb bob is influenced by two forces: gravity and the centrifugal force due to the Earth's rotation. The gravitational force pulls the plumb bob toward the Earth's center, while the centrifugal force caused by the Earth's rotation pushes it away from the Earth's center but along the equatorial plane.
02

Find the orientation of the plumb bob

The actual orientation of the plumb bob will be a result of the balance between gravity and centrifugal force. At the given latitude of 55.0° N, these two forces will both have a component acting in the downward direction, and the plumb bob will align itself with the effective gravity vector. Due to the Earth's shape (oblate spheroid), the actual orientation of gravitational pull is not towards the Earth's center but towards the center of mass of a differential volume, which is closer to the Earth's surface at this latitude.
03

Determine the direction of the centrifugal force

The centrifugal force acts along the same plane as the Earth's rotation, which can be thought of as an imaginary horizontal equatorial plane. At the given latitude, this plane will be tilted with respect to the local vertical direction, which is defined by the effective gravity vector.
04

Analyze the intersection of the line with Earth's axis of rotation

A line extended from the string supporting the plumb bob will not pass through the Earth's center, because it is aligned with the effective gravity vector, which deviates from the line towards the Earth's center due to the centrifugal force acting on the plumb bob. The closer we are to the poles, the more gravity and the lesser centrifugal force influence the plumb bob, and it consequently tends toward the Earth's axis of rotation. As the plumb bob hangs north of the equator (at 55.0° N), it is closer to the North Pole. We can thus conclude that the line from the string supporting the bob, when extended, would intersect the Earth's axis of rotation south of the Earth's center.

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Most popular questions from this chapter

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