# Appendix E:

Newton's Laws,

Pressure-Gradient Force,

and Coriolis Force

# Newton’s Laws of Motion

Because air is composed of atoms and molecules, its motion is governed by the same natural laws that apply to all matter. Simply put, when a force is applied to air, air will be displaced from its original position. Depending on the direction from which the force is applied, the air may move horizontally to produce winds or, in some situations, vertically to generate convective flow. To better understand the forces that produce global winds, it is helpful to become familiar with Newton’s first two laws of motion.

Newton’s first law of motion states that an object at rest will remain at rest, and an object in motion will continue moving at a uniform speed and in a straight line unless a force is exerted upon it. In simple terms, this law states that objects at rest tend to stay at rest, and objects in motion tend to continue moving at the same rate in the same direction. The tendency of things to resist change in motion (including a change in direction) is known as inertia.

You have experienced Newton’s first law if you have ever pushed a stalled auto along flat terrain. To start the automobile moving (accelerating) requires a force sufficient to overcome its inertia (resistance to change). However, once this vehicle is moving, a force equal to that of the frictional force between the tires and the pavement is enough to keep it moving.

Moving objects often deviate from straight paths or come to rest, whereas objects at rest begin to move. The changes in motion we observe in daily life are the result of one or more applied forces.

Newton’s second law of motion describes the relationship between the forces that are exerted on objects and the observed accelerations that result. Newton’s second law states that the acceleration of an object is directly proportional to the net force acting on that body and inversely proportional to the mass of the body. The first part of Newton’s second law means that the acceleration of an object changes as the intensity of the applied force changes.

We define acceleration as the rate of change in velocity. Because velocity describes both the speed and direction of a moving body, velocity can be changed by changing the body’s speed, or its direction, or both. Further, the term acceleration refers both to decreases and increases in velocity.

For example, we know that when we push down on the gas pedal of an automobile, we experience a positive acceleration (increase in velocity). In contrast, using the brakes retards acceleration (decreases velocity).

In the atmosphere, three forces are responsible for changing the state of motion of winds. These are the pressure-gradient force, the Coriolis force, and friction. From the preceding discussion, it should be clear that the relative strengths of these forces will determine to a large degree the role of each in establishing the flow of air. Further, these forces can be directed in such a way as to increase the speed of airflow, decrease the speed of airflow, or, in many instances, just change the direction of airflow.

# Pressure-Gradient Force

The magnitude of the pressure-gradient force is a function of the pressure difference between two points and air density. It can be expressed as

Let us consider an example where the pressure 5 kilometers above Little Rock, Arkansas, is 540 millibars, and at 5 kilometers above St. Louis, Missouri, it is 530 millibars. The distance between the two cities is 450 kilometers, and the air density at 5 kilometers is 0.75 kilogram per cubic meter. To use the pressure-gradient equation, we must use compatible units. We must first convert pressure from millibars to pascals, another measure of pressure that has units of (kilograms × meters−1 × second2).

In our example, the pressure difference above the two cities is 10 millibars, or 1000 pascals (1000 kg/m·s−2). Thus, we have:

Newton’s second law states that force equals mass times acceleration (F = m × a). In our example, we have considered pressure-gradient force per unit mass; therefore, our result is an acceleration (F/m = a). Because of the small units shown, pressure-gradient acceleration is often expressed as centimeters per second squared. In this example, we have 0.296 cm/s2.

# Coriolis Force as a Function of Wind Speed and Latitude

Figure 6.15 shows how wind speed and latitude conspire to affect the Coriolis force. Consider a west wind at four different latitudes (0°, 20°, 40°, and 60°). After several hours, Earth’s rotation has changed the orientation of latitude and longitude of all locations except the equator, such that the wind appears to be deflected to the right. The degree of deflection for a given wind speed increases with latitude because the orientation of latitude and longitude lines changes more at higher latitudes. The degree of deflection of a given latitude increases with wind speed because greater distances are covered in the period of time considered.

We can show mathematically the importance of latitude and wind speed on Coriolis force:

Using this equation, one could calculate the Coriolis force for any latitude or wind speed. Consider Table E.1, which shows the Coriolis force per unit mass for three specific wind speeds at various latitudes. All values are expressed in centimeters per second squared (cm/s−2). Because pressure-gradient force and Coriolis force approximately balance under geostrophic conditions, we can see from our table that the pressure-gradient force (per unit mass) of 0.296 cm/s−2 illustrated in the preceding discussion of pressure-gradient force would produce relatively strong winds.

Table E.1

Coriolis Force for Three Wind Speeds at Various Latitudes