Laws of Motion
2.0 Newton's second law of motion
2.0 Newton's second law of motion
The rate of change of momentum of a body is proportional to the applied force and takes place in the direction in which the force acts.
$$\overrightarrow F = \frac{{d\overrightarrow p }}{{dt}}$$ We also know, $$\overrightarrow p = m\overrightarrow v $$ So, $$\begin{equation} \begin{aligned} \overrightarrow F = \frac{{d\left( {m\overrightarrow v } \right)}}{{dt}} \\ {\overrightarrow F _{net}} = m\frac{{d\overrightarrow v }}{{dt}} + \overrightarrow v \frac{{dm}}{{dt}} \\\end{aligned} \end{equation} $$
The above equation is only valid when mass $m$ and velocity $v$ of body both varies.
Case I: When mass is constant $\left( {\frac{{dm}}{{dt}} = 0} \right)$
$$\begin{equation} \begin{aligned} {\overrightarrow F _{net}} = m\frac{{d\overrightarrow v }}{{dt}} + \overrightarrow v \times 0 \\ {\overrightarrow F _{net}} = m\overrightarrow a \quad \left( {\frac{{d\overrightarrow v }}{{dt}} = \overrightarrow a } \right) \\\end{aligned} \end{equation} $$
Case II: When velocity is constant $\left( {\frac{{d\overrightarrow v }}{{dt}} = 0} \right)$
$$\begin{equation} \begin{aligned} \overrightarrow F = m \times 0 + \overrightarrow v \left( {\frac{{dm}}{{dt}}} \right) \\ \overrightarrow F = \overrightarrow v \frac{{dm}}{{dt}} \\\end{aligned} \end{equation} $$
Note:
1. The second law of motion is strictly applicable for point mass.
2. Second law of motion is applicable in inertial as well as non-inertial frame.
3. If observation is from inertial frame of reference then resultant force is due to real forces only which is proportional to the rate of change in linear momentum.
4. If observation is from non-inertial frame of reference then resultant force is due to real as well as pseudo forces which is proportional to rate of change in linear momentum.
5. The second law of motion is a vector law. It is a combination of three vectors in three mutually perpendicular directions.
$$\begin{equation} \begin{aligned} {\overrightarrow F _{net}} = {\overrightarrow F _x} + {\overrightarrow F _y} + {\overrightarrow F _z} \\ \left| {{{\overrightarrow F }_{net}}} \right| = \sqrt {{F_x}^2 + {F_y}^2 + {F_z}^2} \\\end{aligned} \end{equation} $$
where,
$\begin{equation} \begin{aligned} {\overrightarrow F _x} = m{\overrightarrow a _x} = \frac{{d{p_x}}}{{dt}} \\ {\overrightarrow F _y} = m{\overrightarrow a _y} = \frac{{d{p_y}}}{{dt}} \\ {\overrightarrow F _z} = m{\overrightarrow a _z} = \frac{{d{p_z}}}{{dt}} \\\end{aligned} \end{equation} $
${\overrightarrow a _x}$, ${\overrightarrow a _y}$ and ${\overrightarrow a _z}$ are the acceleration along $x$, $y$ and $z$ directions respectively.
${\overrightarrow p _x}$, ${\overrightarrow p _y}$ and ${\overrightarrow p _z}$ are the momentum along $x$, $y$ and $z$ directions respectively.
${\overrightarrow F _x}$, ${\overrightarrow F _y}$ and ${\overrightarrow F _z}$ are the force along $x$, $y$ and $z$ directions respectively.
