Newton’s Second Law of Motion

Newton’s second law of motion states that the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of the applied force.

Explanation

Newton’s second law of motion means that when a bigger force is applied to a body, its linear momentum changes faster and vice-versa. The momentum will change in the direction of the applied force. To understand it further, suppose

• $m =$ mass of a body,
• $\vec{v} =$ velocity of the body

The linear momentum of the body

$ \vec{p} = m\vec{v} \quad \text{…(1)} $

Let $\vec{F}$ = external force applied on the body in the direction of motion of the body.

$d\vec{p}$ = a small change in linear momentum of the body in a small time $dt$

Rate of change of linear momentum of the body $ \displaystyle =\dfrac{{d\vec{p}}}{{dt}}$

According to Newton’s second law,

$ \displaystyle \dfrac{{d\vec{p}}}{{dt}}\propto \vec{F}\quad \text{or}\quad \vec{F}=k\dfrac{{d\vec{p}}}{{dt}}\quad …\text{(2)}$

where $k$ is a constant of proportionality.

Using (1),

$ \vec{F} = k \dfrac{d}{dt}(m\vec{v}) = k m \dfrac{d\vec{v}}{dt} $

But $ \vec{a} = \dfrac{d\vec{v}}{dt} $ represents acceleration of the body.

$ \therefore \vec{F} = k m \vec{a} \quad \text{…(3)} $

The value of the constant of proportionality $k$ depends on the units adopted for measuring the force. Both in SI and c.g.s. systems, the unit of force is so chosen in such a manner that it makes $k = 1$.

Putting this value of $k$ in (3), we get

$ \displaystyle \vec{F}=m\vec{a}\text{ }…\text{ (4)}$

If the acceleration produced is in the X-axis, having components $a_x, a_y, a_z$ along the X-axis, Y-axis, and Z-axis respectively, then

$ \displaystyle \begin{array}{l}\vec{a}=({{a}_{x}}\hat{i}+{{a}_{y}}\hat{j}+{{a}_{z}}\hat{k})\text{ }…\text{ (5)}\\\vec{F}=m\vec{a}=m({{a}_{x}}\hat{i}+{{a}_{y}}\hat{j}+{{a}_{z}}\hat{k})\text{ }…\text{ (6)}\end{array}$

If $ \vec{F}_x, \vec{F}_y, \vec{F}_z $ are the components of $\vec{F}$ along the X, Y, Z axes respectively, then

$ \displaystyle \vec{F}={{{\vec{F}}}_{x}}\hat{i}+{{{\vec{F}}}_{y}}\hat{j}+{{{\vec{F}}}_{z}}\hat{k}\text{ }…\text{ (7)}$

Comparing (6) and (7)

$ \displaystyle {{{\vec{F}}}_{x}}=m{{a}_{x}},\quad {{{\vec{F}}}_{y}}=m{{a}_{y}},\quad {{{\vec{F}}}_{z}}=m{{a}_{z}}\text{ }…\text{ (8)}$

As acceleration is a vector quantity and so is force, therefore $ \vec{F} $ being the product of $m$ and $ \vec{a} $, is a vector.

The direction of $ \vec{F} $ is the same as the direction of $ \vec{a} $.

Note that the acceleration component along a given axis can be calculated by taking the force components along the same axis, and dividing it by the mass of the body.

Equation (4) represents the equation of motion of a body.

We can rewrite equation (4) in scalar form as

$ F = m a $

Thus the magnitude of force can be calculated by multiplying mass of the body with the acceleration produced in it. Hence, the second law of motion gives us a measure of force.