Question

The Coriolis force can produce a torque on a spinning object. To illustrate this, consider a horizontal hoop of mass \(m\) and radius \(r\) spinning with angular velocity \(\omega\) about its vertical axis at colatitude \(\theta\). Show that the Coriolis force due to the earth's rotation produces a torque of magnitude \(m \omega \Omega r^{2} \sin \theta\) directed to the west, where \(\Omega\) is the earth's angular velocity. This torque is the basis of the gyrocompass.

Short answer

The torque due to the Coriolis force is \( m \omega \Omega r^2 \sin \theta \) westward.

Step-by-step solution

Understanding the Coriolis Force

The Coriolis force is an inertial force that acts on objects moving within a rotating reference frame. It is defined by the expression: \( F_c = -2m (\vec{\Omega} \times \vec{v}) \), where \(\vec{\Omega}\) is the angular velocity of the Earth's rotation and \(\vec{v}\) is the velocity of the moving object.

Calculate Velocity Vector

As the hoop spins with an angular velocity \( \omega \), a point on the hoop at a distance \( r \) from the center moves with velocity \( \vec{v} = \vec{r} \times \vec{\omega} \). In simplified form, due to circular motion in a horizontal plane, the velocity of any point on the hoop is given by \( v = r\omega \).

Identify the Coriolis Force

Substitute the velocity into the Coriolis force equation. We have \( F_c = -2m (\vec{\Omega} \times \vec{v}) \). The cross product yields a force component perpendicular to both the velocity \( \vec{v} \) and the Earth's rotation vector \( \vec{\Omega} \).

Simplify the Cross Product

Given that \( \vec{\Omega} \) is the Earth's rotational angular velocity vector pointing north and \( \vec{v} \) is directed tangentially due to the hoop's rotation, their cross product in magnitude becomes \( |\vec{\Omega} \times \vec{v}| = \Omega v \sin \theta = \Omega r \omega \sin \theta \).

Determine Torque Direction and Magnitude

The Coriolis force acts in a direction causing a torque around the vertical axis of the hoop. The magnitude of this torque due to a tangential force is given by \( \tau = rF_c = m \omega \Omega r^2 \sin \theta \). The direction of the torque is such that it is directed toward the west due to the right-hand rule and the direction of rotation.

Key concepts

Coriolis effect

The Coriolis effect is a fascinating phenomenon that occurs in rotating systems, like our very own Earth. It arises because of the Coriolis force, which is an inertial force acting on objects in motion within a rotating reference frame. Imagine you are on a merry-go-round and you toss a ball to a friend sitting on the opposite side. Even though you aim straight, your friend sees the ball curve. This is due to the Coriolis effect.

On Earth, as the planet rotates, this effect influences the path of moving objects like air currents, ocean currents, and even tiny particles. This force doesn't change the speed of the object but alters its direction of motion. The Coriolis force is mathematically described by the equation: \[F_c = -2m (\vec{\Omega} \times \vec{v})\], where \(\vec{\Omega}\) is the angular velocity vector of Earth, and \(\vec{v}\) is the velocity of the moving object.

In our exercise, this effect helps create a torque on a spinning hoop, illustrating how the Coriolis force causes rotation itself to produce additional rotational effects.

torque

Torque is a measure of the force that causes an object to rotate. You can think of it as the rotational equivalent of linear force. Imagine turning a wrench: when you apply force at a handle's end, you exert torque, causing the bolt to turn. The torque's magnitude depends on the force applied, the distance from the axis of rotation, and the angle at which the force is applied.

In mathematical terms, torque \( \tau \) is calculated by: \[\tau = rF \sin \theta\], where \(r\) is the distance from the pivot point, \(F\) is the applied force, and \(\theta\) is the angle the force makes with the line from the pivot.

In the context of our spinning hoop, the Coriolis force creates a torque that is proportional to the mass \(m\), the hoop's angular velocity \(\omega\), Earth's angular velocity \(\Omega\), and the radius squared \(r^2\), specifically along the hoop's rotation direction. This is the basis of the gyrocompass, guiding vessels by aligning them with Earth's axis rotation.

angular velocity

Angular velocity is a vector quantity that describes how fast something rotates and in which direction. It provides both the rate of rotation and the axis around which the object rotates. Imagine watching a fan spin: the blades rotate around a fixed point at a specific speed, which is quantified by the angular velocity.

Mathematically, angular velocity \( \omega \) is expressed in radians per second. It can be derived using \( \omega = \frac{d\theta}{dt}\), where \(\theta\) is the angle the object has rotated, and \(t\) is time.

In our spinning hoop problem, the hoop rotates with angular velocity \(\omega\) about a vertical axis. This spin is subject to another angular velocity, \(\Omega\), from Earth's rotation. Thus, multiple angular velocities interact, contributing to complex motions due to the Coriolis effect. Here, the combined rotations result in measurable torques applied to the hoop.

rotating reference frame

A rotating reference frame is a perspective that rotates with the object or system being considered. It's like being on a carousel: you see everything relative to the carousel's rotation, leading to phenomena not present in a non-rotating frame.

Objects in rotating reference frames experience fictitious forces, like the Coriolis force, which appear due to the rotation. These forces are not observed in an inertial, non-rotating frame. It's essential to understand rotating frames for analyzing systems affected by Earth's spin, like weather patterns or gyroscopic devices.

In our exercise, the hoop is in a rotating reference frame because it spins as Earth rotates. The Coriolis force acts in this frame, producing a torque. This frame enables us to understand behaviors that wouldn't be apparent purely by observations made from a static perspective, and are vital for applications spanning from meteorology to navigation and beyond.