By Dr. Shahid Ali Yousafzai

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Actually whenever there will be an acceleration, the net forces should not be zero.

Second misconception about the Newton’ s 2nd law of motion is that we have forces on both sides of the equation but we have force on one side and acceleration on other side as mass is constant. Acceleration can never be a force. We can resolve this issue by considering the concept of inertia.

The Newton’ s law is valid only in inertial frame of reference and not the non-inertial frame of references.

Let’ s start with the Hooke’ s law $$ F = -kx $$ The very first thing is this is not the Hooke’ s law; this is also a kind of relation. The Hooke’ s law is actually defined as: $$ \text { Elastic Modulus } = \frac { Stress }{ Strain } $$ The relation between stress and strain depends upon the material. Elastic material restores their original position while there are some materials that do not restore. So if it’ s a law, it would have everything incorporated in it.

In the plot, we have for a certain strain, a linear relation between stress and strain. Beyond that range, the relation is nonlinear. We will restrict ourselves in F = -kx to only the linear range.

Which one is dependent and which one is independent here? Force is independent and x is dependent. So, $$ x \propto F $$ and then $$ x = constant \times F $$ What should be the constant? Now come again to the plot.

Let’ s say you are having a spring and you attach some mass of 1kg to it. This mass will produce some displacement in this spring. The displacement is supposed to be extremely small. The constant will define me the stiffness of the spring.

For 1N of force, 1m displacement is produced. 1m is huge compared to 1N. This constant should adjust that thing which means it should be less than a meter or we should give it a value that is less than 1. So we are in the range where 1N is not producing 1m displacement. Thus, the constant should be 1/k. Then, $$ F = kx $$ As, the spring goes back and forth around its mean position, so I will write a minus sign. $$ F = -kx $$ You can have many other linear proportionality laws like this, for example $$ \vec { P } ={ \varepsilon }_{ 0 }{ \chi }_{ e }\vec { E } $$ And many more. Bring them to the straight line terminologies and see what the constants should be.

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