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We will discuss the very fundamental laws of Physics especially those laws of physics which are having linear proportionalities and we will discuss the physics and the concept behind them, the different misconceptions about them and to understand all those, we will have to understand the straight line equation.
So, the very first one that I am writing is, for example, the very first law, which is Ohm’s law. Now you know Ohm’s law is written as $$ V = IR $$ We will discuss this thing after a while that whether it’s a law or not but first we are discussing its equation. If I write this as $$ V\propto I $$and then $$ V=Constant\times I $$ The constant of proportionality is (R), so this will make our equation $$ V = IR $$
Another way to do this thing is if I write that $$ I\propto V $$ Then $$ I = Constant \times V $$ and this constant I write as $$ I=\frac { 1 }{ R } V $$ So this will make again this equation $$ V=IR $$ Now the question is, whether we should write it like $$ V= constant \times I =IR $$ Or $$I = constant \times V=\frac { 1 }{ R } V \Rightarrow V =IR $$What is the correct way? This we will have to decide so in the beginning I am saying that it is not right to write it like the first but to write it like the second and I will explain you the reason of the concept behind this.
Similarly if I write the Newton’s 2nd law of motion, we usually write as $$ \vec { F } =m\vec { a } $$ Now if I write in magnitude form then this is $$ F=ma $$Now how to get this equation? The very first thing that we will have to say about this one should be like $$F \propto a $$ and then $$ F =constant \times a $$ and then this constant is the mass and this is equal to $$ F = ma $$ Or we should $$ a \propto F $$ then $$ a=constant \times F $$ and then $$a=\frac { 1 }{ m } F\Rightarrow F=ma $$There is another way that I write $$ a\propto F $$ and $$ a \propto \frac { 1 }{ m } $$ so I can combine the two proportionalities and I can write $$ a \propto \frac { F }{ m } $$ and then I write that $$ a =constant \frac { F }{ m } $$ and this constant is 1 $$ F =ma $$ What is the right equation among all this? And then you know about the Hook’s law. The Hook’s law as you know is $$ \vec { F } =-k\vec { x } $$ where the - sign is showing the restoration nature of this force. So, if I write in magnitude form $$ F =-kx $$ Now how to write this one like whether we should write $$ F \propto -x $$ and then $$F = Constant \times (-x) = -kx$$ where \(k\) is the constant of proportionality.
Or we should write this thing as $$ x \propto F $$then $$x =constant \times F $$ and then $$ x=\frac { 1 }{ k } F $$ and this is $$ F =kx $$ Which way is right? Just put a - sign to show the restoration nature of this force. $$ F=-kx $$ All these three laws are actually having one thing in common and that thing is that they are linear laws. The equation is a linear equation.
We will discuss them one by one but the very first thing is that we will have to understand a linear equation.
And a linear equation you know is the equation of a straight line. So the very first thing that we would like to discuss is to discuss the equation of straight line. $$ y=mx+c $$
and now I will have to split it and discuss individually all of these parameters.
We call \(x\) as the independent variable and \(y\) as a dependent variable, while \(c\) is the y-intercept and \(m\) is called the slope.
This \(m\) is defined as slope and this is actually the rise over run. How much we are rising along the run.
The plot will be $$ y=mx+c $$ \(y\) will be the output and \(x\) is the input. We will give value to \(x\) and we will get value of \(y\).
Whatever you plot on x-axis is actually the \(x\) value and whatever you plot on this y-axis is the \(y\) value.
Whatever you plot in the plane, which is the xy-plane is actually the value you call \( \frac { y }{ x } \)
This is not \(y+x\) or \(y-x\).
This value whatever you plot here, this point is actually \( \frac { y }{ x } \) and this is called the slope.
Now if I will have an equation and the equation is like let’s say the very first equation that I say $$y=x$$ Now what is here? Here the value \(m=1\) and the value \(c=0\).
So for \(x=1\), \(y=1\). For \(x=2\), \(y=2\) and so on. In all these cases $$ m=\frac { y }{ x } $$ and it will always come out to be 1.
So, we will have a straight line and the straight line will be somehow exactly at \(45^\circ \) angle. because \(x\) and \(y\) should be equal.
Now what the intercept will actually do here? If I now write that$$y=x+1$$ Now in this equation, \(m=1\) but \(c\) is also 1.
Whatever you do with this equation it will be the equation of a straight line because \(y\) is in linear relation with \(x\).
If the \(x\) power will change, some curvature will occur here. But as long as the \(x\) power is 1 and the y power is 1, it will be always a straight line.
So, I can say that for this one, If I consider \(x=0\) point, then \(y=1\).
How much you extend this line, the slope will remain the same. The slope of this line is the same as this line because it is also making \(45^{\circ}\). The only difference is this the y-offset or the y- intercept has changed.
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