equations:newtons_second_law
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
equations:newtons_second_law [2018/03/30 11:10] jakobadmin [Intuitive] |
equations:newtons_second_law [2018/12/15 12:01] (current) 93.132.16.208 Small change in the previous edit |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | <WRAP lag>$\color{blue}{\vec F}=\color{olive}{m}\color{magenta}{\vec a}$</WRAP> | + | <WRAP lag>$\color{blue}{\vec F}(\color{red}{\vec r})=\color{olive}{m}\color{magenta}{\vec a}$</WRAP> |
| ====== Newton's Second Law ====== | ====== Newton's Second Law ====== | ||
| Line 5: | Line 5: | ||
| <tabbox Intuitive> | <tabbox 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. | + | 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 divided by its $\color{olive}{\text{mass}}$. | + | 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 | 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 | ||
| - | * [[equations:newtons_law|Newton's law of gravity]] | + | * [[formulas:newtons_law|Newton's law of gravity]] |
| - | * [[equations:lorentz_force_law|Lorentz' force law]] | + | * [[formulas:lorentz_force_law|Lorentz' force law]] |
| - | * [[equations:coulombs_law|Coulomb's force law]] | + | * [[formulas:coulombs_law|Coulomb's force law]] |
| ---- | ---- | ||
| - | 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 [[equations:newtons_law|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. | + | 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 [[formulas:newtons_law|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. |
| Line 24: | Line 26: | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
| - | <note tip> | + | |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | **Boundary or Initial Conditions** |
| - | </note> | + | |
| - | + | 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 [[formulas:newtons_law|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 [[https://math.stackexchange.com/questions/1781/can-this-gravitational-field-differential-equation-be-solved-or-does-it-not-sho|here]]. | ||
| + | |||
| + | Take note that before we can fully solve the equation, we need to specify the [[basic_notions:boundary_conditions|boundary conditions]]. | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| <tabbox Abstract> | <tabbox Abstract> | ||
| Line 34: | Line 73: | ||
| </note> | </note> | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Why is it interesting?> |
| + | |||
| + | |||
| + | Newton's second law is the most fundamental equation of [[theories:classical_mechanics:newtonian|classical mechanics]]. It is still used nowadays, for example, by engineers. | ||
| + | |||
| + | ---- | ||
| - | Newton's second law is the most fundamental equation of [[theories:classical_mechanics|classical mechanics]]. It is still used nowadays, for example, by engineers. | + | <blockquote>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 $F=ma$.<cite>[[https://insidetheperimeter.ca/myfavouriteequation-scientists-share-their-fave-formulae/|Rob Moore]]</cite></blockquote> |
| <tabbox Definitions> | <tabbox Definitions> | ||
equations/newtons_second_law.1522401009.txt.gz · Last modified: 2018/03/30 09:10 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
