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. axis of rotation south or north of the Earth's center?

Short answer

Answer: The Earth's center is north of the plumb bob at a latitude of 55.0°N.

Step-by-step solution

Understand the forces acting on the plumb bob

The plumb bob hangs motionlessly with respect to the ground beneath it. There are two forces acting on the plumb bob: gravity, which pulls it towards the Earth's center, and the centrifugal force due to Earth's rotation. The plumb bob creates an equilibrium between these forces and hangs under a slight angle.

Determine the direction of the centrifugal force and its effect on the bob

The centrifugal force is an outward force acting on objects in a rotating frame of reference, like the Earth. In this case, the centrifugal force acts at a right angle to Earth's axis of rotation. At a latitude of \(55.0^{\circ} \mathrm{N}\), the centrifugal force acts horizontally towards the equator. As the plumb bob hangs at an angle, the centrifugal force acts perpendicular to the gravity force, and the string should turn in a direction opposite to the centrifugal force.

Find out if the Earth's center is north or south of the plumb bob

Since the centrifugal force is acting in the equator's direction and the plumb bob hangs at an angle opposite to this force, it implies that the Earth's center (where gravity pulls to) is slightly north of the bob. This causes the line passing through the string not to pass exactly through Earth's center. Hence, the Earth's center is north of the plumb bob. The axis of rotation is south of the Earth's center in this exercise, as the centrifugal force pushes the bob slightly northward, and the Earth's center is found to be north of the plumb bob.

Key concepts

Centrifugal Force and Its Role in Physics

The concept of centrifugal force plays a significant role in understanding various phenomena in physics, particularly when analyzing forces within rotating frames of reference. When we refer to centrifugal force, we are describing the apparent force that seems to push objects away from the center of rotation. In everyday life, you may have experienced this as the force pressing you against the door of a car while it makes a sharp turn. Technically, it's an inertial force, meaning it arises due to the inertia of an object's motion rather than a physical interaction with another object.

Now, consider the Earth, which is a rotating sphere. Those living on its surface are in a constant rotating frame of reference. Because of the Earth's rotation, any object attached to the Earth, including a plumb bob, experiences a centrifugal force that pushes it outward, away from the axis of rotation. This centrifugal force is responsible for the small but measurable effect of a plumb bob not hanging exactly in line with the gravitational pull towards the center of the Earth. Instead, it hangs at a slight angle pointing away from the axis, showing the balance between the gravitational and centrifugal forces.

Earth's Rotation and Its Effects

Earth's rotation refers to the planet's spinning on its axis. This rotation has significant effects on the world, including day and night cycles, weather patterns, and even subtle effects on the movement of objects. When considering the rotation in terms of physics, it's vital to understand how it alters our perception and experience of forces. For instance, the rotation of Earth affects the direction in which things fall—technically, the plumb bob. Instead of falling in what we'd perceive as a 'straight' line towards the Earth's center, the rotation causes the bob to be slightly deflected due to the centrifugal force. This is particularly evident when considering the plumb bob's behavior at different latitudes; it does not hang in a line that intersects exactly with the Earth's center. This phenomenon helps scientists understand and measure the Earth’s rotation and contributes to setting up precise geodetic measurements for maps and satellite technology.

Latitude Effects on Gravity

Gravity is often thought of as a uniform force pulling everything directly towards the center of the Earth; however, this is an oversimplification. One of the intriguing ways that gravity varies is by latitude, thanks to the Earth's rotation and its shape. Because the Earth is not a perfect sphere but an oblate spheroid (wider at the equator and flatter at the poles), the gravitational pull is stronger at the poles and weaker at the equator. This is due to both the distance from the center of Earth—shorter at the poles—and the centrifugal force, which is strongest at the equator, counteracting gravity to some degree.

For a plumb bob at latitude 55.0° N, the combination of Earth's gravitational pull and the centrifugal force caused by Earth's rotation means the bob does not hang in a perfect vertical line in relation to its surroundings. Instead, its exact hanging direction will be towards the center of the Earth but will also reflect the influence of the centrifugal force and the Earth's shape, tilting very slightly away from the pole and toward the equator. Understanding these subtleties allows scientists and engineers to compensate for such effects in precision tasks like surveying and the creation of accurate models of Earth's gravitational field.