10.1: Newton's 2nd Law
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Isaac Newton’s published his laws in Latin, the language of natural philosophy (science) at the time (1687). Here is the translation from Newton’s Philosophiæ Naturalis Principia Mathematica (“Mathematical Principles of Natural Philosophy”):
“Law I. Every body perseveres in its state of being at rest or of moving uniformly straight forward, except inasmuch as it is compelled by impressed forces to change its state.
“Law II. Change in motion is proportional to the motive force impressed and takes place following the straight line along which that force is impressed.
“Law III. To any action, there is always a contrary, equal reaction; in other words, the actions of two bodies each upon the other are always equal and opposite in direction.
“Corollary 1. A body under the joint action of forces traverses the diagonal of a parallelogram in the same time as it describes the sides under their separate actions.”
10.2.1. Lagrangian
For a Lagrangian framework (where the coordinate system follows the moving object), Newton’s Second Law of Motion is
\(\ \begin{align} \vec{F}=m \cdot \vec{a}\tag{10.1}\end{align}\)
where \(\ \vec{F}\) is a force vector, m is mass of the object, and \(\ \vec{a}\) is the acceleration vector of the object. Namely, the object accelerates in the direction of the applied force.
Acceleration is the velocity \(\ \vec{V}\) change during a short time interval ∆t:
\(\ \begin{align} \vec{a}=\frac{\Delta \vec{V}}{\Delta t}\tag{10.2}\end{align}\)
Plugging eq. (10.2) into (10.1) gives:
\(\ \begin{align} \vec{F}=m \cdot \frac{\Delta \vec{V}}{\Delta t}\tag{10.3a}\end{align}\)
Recall that momentum is defined as \(m \cdot \vec{V}\). Thus, if the object’s mass is constant, you can rewrite Newton’s 2nd Law as Lagrangian momentum budget:
\(\ \begin{align} \vec{F}=\frac{\Delta(m \cdot \vec{V})}{\Delta t}\tag{10.3b}\end{align}\)
Namely, this equation allows you to forecast the rate of change of the object’s momentum.
If the object is a collection of air molecules moving together as an air parcel, then eq. (10.3a) allows you to forecast the movement of the air (i.e., the wind). Often many forces act simultaneously on an air parcel, so we should rewrite eq. (10.3a) in terms of the net force:
\(\ \begin{align} \frac{\Delta \vec{V}}{\Delta t}=\frac{\vec{F}_{n e t}}{m}\tag{10.4}\end{align}\)
where \(\vec{F}_{n e t}\) is the vector sum of all applied forces, as · a given by Newton’s Corollary 1 (see the INFO box).
For situations where \(\vec{F}_{n e t} / m=0\), eq. (10.4) tells us that the flow will maintain constant velocity due to inertia. Namely, \(\Delta \vec{V} / \Delta \mathrm{t}=0\) implies that \(\vec{V}\) = constant (not necessarily that \(\vec{V}\) = 0).
Sample Application
If a 1200 kg car accelerates from 0 to 100 km h–1 in 7 s, heading north, then: (a) What is its average acceleration? (b) What vector force caused this acceleration?
Find the Answer
Given: \(\vec{V}_{\text {initial }}=0\), \(\vec{V}_{\text {final }}\) = 100 km h–1 = 27.8 m s–1, tinitial = 0, tfinal = 7 s. Direction is north. m = 1200 kg.
Find: (a) \(\vec{a}\) = ? m·s–2, (b) \(\vec{F}\) = ? N
(a) Apply eq. (10.2): \(\vec{a}=\frac{\Delta \vec{V}}{\Delta t}\)
= (27.8 – 0 m s–1) / (7 – 0 s) = 3.97 m·s–2 to the north
(b) Apply eq. (10.1): \(\vec{F}\) = (1200 kg) · ( 3.97 m·s –2 )
= 4766 N to the north
where 1 N = 1 kg·m·s–2 (see Appendix A).
Check: Physics and units are reasonable.
Exposition: My small car can accelerate from 0 to 100 km in 20 seconds, if I am lucky. Greater acceleration consumes more fuel, so to save fuel and money, you should accelerate more slowly when you drive.
In Chapter 1 we defined the (U, V, W) wind components in the (x, y, z) coordinate directions (positive toward the East, North, and up). Thus, we can split eq. (10.4) into separate scalar (i.e., non-vector) equations for each wind component:
\( \begin{align} \frac{\Delta U}{\Delta t}=\frac{F_{x\ n e t}}{m}\tag{10.5a}\end{align}\)
\(\ \begin{align} \frac{\Delta V}{\Delta t}=\frac{F_{y\ n e t}}{m}\tag{10.5b}\end{align}\)
\(\ \begin{align} \frac{\Delta W}{\Delta t}=\frac{F_{z\ n e t}}{m}\tag{10.5c}\end{align}\)
where Fx net is the sum of the x-component of all the applied forces, and similar for Fy net and Fz net .
From the definition of ∆ = final – initial, you can expand ∆U/∆t to be [U(t+∆t) – U(t)]/∆t. With similar expansions for ∆V/∆t and ∆W/∆t, eq. (10.5) becomes
\(\ \begin{align} U(t+\Delta t)=U(t)+\frac{F_{x n e t}}{m} \cdot \Delta t\tag{10.6a}\end{align}\)
\(\ \begin{align} V(t+\Delta t)=V(t)+\frac{F_{y\ n e t}}{m} \cdot \Delta t\tag{10.6b}\end{align}\)
\(\begin{align}W(t+\Delta t)=W(t)+\frac{F_{z\ n e t}}{m} \cdot \Delta t\tag{10.6c}\end{align}\)
These are forecast equations for the wind, and are known as the equations of motion. The Numerical Weather Prediction (NWP) chapter shows how the equations of motion are combined with budget equations for heat, moisture, and mass to forecast the weather.
Sample Application
Initially still air is acted on by force Fy net/m = 5x10–4 m·s–2 . Find the final wind speed after 30 minutes.
Find the Answer
Given: V(0) = 0, Fy net/m = 5x10–4 m·s–2 , ∆t = 1800 s
Find: V(∆t) = ? m s–1. Assume: U = W = 0.
Apply eq. (10.6b): V(t+∆t) = V(t) + ∆t · (Fy net/m)
= 0 + (1800s)·(5x10–4 m·s–2) = 0.9 m s–1.
Check: Physics and units are reasonable.
Exposition: This wind toward the north (i.e., from 180°) is slow. But continued forcing over more time could make it faster.
10.2.2. Eulerian
While Newton’s 2nd Law defines the fundamental dynamics, we cannot use it very easily because it requires a coordinate system that moves with the air. Instead, we want to apply it to a fixed location (i.e., an Eulerian framework), such as over your house. The only change needed is to include a new term called advection along with the other forces, when computing the net force Fnet in each direction. All these forces are explained in the next section.
But knowing the forces, we need additional information to use eqs. (10.6) — we need the initial winds [U(t), V(t), W(t)] to use for the first terms on the right side of eqs. (10.6). Hence, to make numerical weather forecasts, we must first observe the current weather and create an analysis of it. This corresponds to an initial-value problem in mathematics.
Average horizontal winds are often 100 times stronger than vertical winds, except in thunderstorms and near mountains. We will focus on horizontal forces and winds first.
As a child at Woolsthorpe, his mother’s farm in England, Isaac Newton built clocks, sundials, and model windmills. He was an average student, but his schoolmaster thought Isaac had potential, and recommended that he attend university.
Isaac started Cambridge University in 1661. He was 18 years old, and needed to work at odd jobs to pay for his schooling. Just before the plague hit in 1665, he graduated with a B.A. But the plague was spreading quickly, and within 3 months had killed 10% of London residents. So Cambridge University was closed for 18 months, and all the students were sent home.
While isolated at his mother’s farm, he continued his scientific studies independently. This included much of the foundation work on the laws of motion, including the co-invention of calculus and the explanation of gravitational force. To test his laws of motion, he built his own telescope to study the motion of planets. But while trying to improve his telescope, he made significant advances in optics, and invented the reflecting telescope. He was 23 - 24 years old.
It is often the young women and men who are most creative — in the sciences as well as the arts. Enhancing this creativity is the fact that these young people have not yet been overly swayed (perhaps misguided) in their thinking by the works of others. Thus, they are free to experiment and make their own mistakes and discoveries.
You have an opportunity to be creative. Be wary of building on the works of others, because subconsciously you will be steered in their same direction of thought. Instead, I encourage you to be brave, and explore novel, radical ideas.
This recommendation may seem paradoxical. You are reading my book describing the meteorological advances of others, yet I discourage you from reading about such advances. You must decide on the best balance for you.