Question

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

Answer: South of the Earth's center.

Step-by-step solution

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.

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.

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.

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.

Key concepts

Gravity

Gravity is a fundamental force that draws objects with mass toward one another. On Earth, it pulls everything, including us, toward its center. This is what makes things "fall" to the ground when we drop them. For a plumb bob, gravity is the force that attempts to pull it directly toward the center of the Earth. However, Earth's shape isn't a perfect sphere; it’s more like a slightly flattened sphere, or an oblate spheroid.

This shape means that gravity does not always point directly toward the center from every point on the surface. Instead, it points toward the center of mass of the Earth. On the outer surface, especially at different latitudes, the actual gravitational pull is slightly tilted compared to a direct line to the Earth's center. This tilt becomes evident when the location is away from the equator, like at 55° N. As you move from the equator towards the poles, gravity's direction adjusts subtly, affecting objects like the plumb bob.

Centrifugal Force

Centrifugal force is an apparent force that acts outward on objects moving around a center, arising from the object’s inertia. Imagine spinning a ball on a string in a circle. The ball seems to pull away from you as it spins. This is similar to what happens on a rotating planet like Earth.

Due to Earth's rotation, part of what you feel as "heaviness" or weight is actually reduced slightly by the centrifugal force. This force pushes objects away from the axis of rotation, which, for Earth, runs from the North Pole to the South Pole. It's strongest at the equator because that is where Earth spins the fastest, and it becomes nearly zero moving towards the poles.

At a latitude like 55° N, centrifugal force is neither at its weakest nor its strongest, but it's significant enough to tilt the plumb bob's alignment slightly away from pointing directly to the Earth's center. This is why the combination of gravity and centrifugal forces determines the bob's orientation.

Earth's Rotation

The Earth's rotation is the spinning of the planet around its axis. It takes about 24 hours for one complete rotation, which gives us our day and night cycle. This rotation affects everything on Earth in various ways, including weather patterns, ocean currents, and the behavior of objects due to inertial forces.

The rotation introduces the centrifugal force mentioned earlier, which alters how gravity is felt. Because of the rotation, true down—the direction a plumb bob would ideally point if only gravity worked alone—is adjusted to an effective down direction. This effective gravity vector is what the plumb bob aligns with, influenced by both the gravitational pull to the center and the outward push of the centrifugal force.

Since the Earth is wider at the equator than at the poles, the speed of rotation varies with latitude. This directly impacts how significant the centrifugal force is compared to gravity at different points on the Earth's surface, making Earth's rotation a critical factor in understanding how forces balance.

Latitude Effect

Latitude effects how the forces of gravity and centrifugal force interact with objects on Earth. Latitudes are imaginary lines circling the globe, parallel to the equator, telling us how far north or south a point is relative to the equator.

At the poles ( 0° or 180° latitude), centrifugal force is minimal because these locations are almost stationary in Earth's rotation. At the equator ( 90° latitude), centrifugal force is at its maximum due to faster rotational speed. When an object is positioned at 55° N, like our exercise's plumb bob, it experiences a balance of these forces differently than at the poles or equator.

This position results in a gravitational pull that doesn't go straight to Earth's center due to the centrifugal force offset. Therefore, the effective direction the plumb bob points is slightly to the north of the line that would intersect through Earth's center from the surface, creating the latitude effect observed in this exercise.