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equations:newtons_second_law

$\color{blue}{\vec F}(\color{red}{\vec r})=\color{olive}{m}\color{magenta}{\vec a}$

# Newton's Second Law

## Intuitive

Newton's second law tells us that how an object $\color{magenta}{\text{gets faster}}$ depends on its $\color{olive}{\text{mass}}$ and the $\color{blue}{\text{total force }}$ acting on it. The total force acting on it depends on the $\color{red}{\text{location of the object.}}$

Formulated a bit differently, it tells us that the $\color{magenta}{\text{acceleration}}$ of an object is given by the ratio of the $\color{blue}{\text{force }}$ acting on it and its $\color{olive}{\text{mass}}$.

The acceleration is the rate of change of the velocity. The velocity is the rate of change of the location.

However, take note that it is not sufficient to describe a physical system. Additionally, to describe a system we need to know what forces act on the object and what equations describe them. Famous examples of such force laws are

So, for example, when we want to describe the movement of a planet around the sun we need to think about what forces act on the plant. For this system gravity is the most important force since both objects - the sun and the planet - a superheavy. Therefore, to calculate the movement of the planet, all we have to do is use Newton's law of gravity to calculate the force acting on it. Then, when we have calculated the force we can use Newton's second law to calculate the acceleration of the object. Then, given some starting point and starting velocity of the planet we can calculate where the planet will be at every point in time in the future.

## Concrete

Boundary or Initial Conditions

Since Newton's second law contains the second derivative of the location: $\vec a = \frac{\partial^2 r}{\partial t^2}$, we need two boundary conditions to solve it. Whenever the independent variable is time, boundary conditions are usually called initial conditions. (In technical terms, we say that Newton's law is a second order differential equation.) For example, we can use the location of the object at the starting time and the velocity at the starting time as boundary conditions. Alternatively, we could use, for example, the location of the object at two different points in time as boundary conditions.

Example: The movement of an asteroid towards the sun

As a first step, we use Newton's law of gravity to calculate the force acting on the asteroid

$$F = G \frac{m_s m_a}{r^2},$$

where $G$ is the gravitational constant, $m_s$ the mass of the sun, $m_a$ the mass of the asteroid and $r$ the distance between them. We assume that the asteroid is much lighter than the sun and therefore can neglect the effect the asteroid has on the sun. In other words, we assume the sun remains stationary, although of course an equal gravitational force also acts on the sun.

We can then put this equation into Newton's second law

$$F = ma \quad \rightarrow \quad G \frac{m_s m_a}{r^2} = ma .$$

Next, we recall that the acceleration $a$ is just the rate of change of the velocity and subsequently that the velocity is just the rate of change of the location:

$$a = \dot v = \ddot r .$$

Putting this into our equation yields a differential equation for the location $r$ that we need to solve

$$G \frac{m_s m_p}{r^2} = m \ddot r .$$

The equation is not easy to solve, and a nice discussion can be found here.

Take note that before we can fully solve the equation, we need to specify the boundary conditions.

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## Why is it interesting?

Newton's second law is the most fundamental equation of classical mechanics. It is still used nowadays, for example, by engineers.

I can't think of another equation that has had more dramatic impact on our history and our world. So much of what we have built around us started with knowing $F=ma$.Rob Moore

## Definitions

• $\vec F$ is the force that acts on the object in question.
• $\vec a$ is the acceleration of the object. The acceleration is the rate of change of the velocity $\vec v$ of the object, which in turn is the rate of change of the location $\vec r$ of the object: $a = \ddot{r} = \dot{v}$. 