1. Statement of the Second Law

Newton’s Second Law states that the net external force acting on an object equals the rate of change of its momentum.

General form:

\(\Sigma{\vec{F}}=\frac{d\vec{p}}{dt}\)​​

where:

• \(\Sigma{\vec{F}}\) = net external force
• \(vec{p}\)​ = momentum

2. Constant Mass Form

For constant mass:

\(\vec{p}=m\vec{v}\)

Substituting into the momentum form:

\(\Sigma\vec{F}=\frac{d(m/vec{v})}{dt}\)​

If mass is constant:

\(\Sigma\vec{F}=m\frac{d(m/vec{v})}{dt}\)​​

Since

\(\vec{a}=\frac{d\vec{v}}{dt}\)​

we obtain the familiar equation:

\(\Sigma\vec{F}=m\vec{a}\)

3. Meaning of the Law

Newton’s Second Law explains how forces change motion.

Key ideas:

• net force causes acceleration
• acceleration points in the direction of net force
• larger force → larger acceleration
• larger mass → smaller acceleration (for the same force)

4. Vector Nature

Force and acceleration are vectors.

Thus Newton’s Second Law applies separately along each axis:

\(\SignaF_x=ma_x\)​

\(\SignaF_y=ma_y\)​

\(\SignaF_z=ma_z\)​

5. Units of Force

SI unit of force:

\(1N=1kg\cdot m/s^2\)

A force of 1 newton produces an acceleration of 1 m/s² on a $1\text{kg}$ object.

6. Free-Body Diagrams

To apply Newton’s Second Law, we draw a free-body diagram (FBD).

An FBD shows:

• the object isolated from surroundings
• all external forces acting on it

Typical forces include:

• gravity
• normal force
• tension
• friction
• applied forces

7. Example: Horizontal Motion

A box of mass $m$ is pushed with force $F$ on a frictionless surface.

Net force:

\(\Sigma{F_x}=F\)

Using Newton’s Second Law:

\(F=ma\)

Acceleration:

\(a=\frac{F}{m}\)​

8. Equilibrium as a Special Case

If the net force is zero:

\(\Sigma\vec{F}=0\)

then:

\(\vec{a}=0\)

This corresponds to Newton’s First Law.

9. Relationship to Momentum

The momentum form:

\(\Sigma{\vec{F}}=\frac{d\vec{p}}{dt}\)​​

is more general and remains valid even when mass changes.

This becomes important in advanced mechanics topics such as rocket motion.

10. Physical Interpretation

Newton’s Second Law provides the direct connection between:

• force
• mass
• acceleration

It explains why:

• heavier objects are harder to accelerate
• stronger forces produce greater changes in motion
• acceleration follows the direction of net force

Summary

General form:

\(\Sigma{\vec{F}}=\frac{d\vec{p}}{dt}\)​​​

Constant-mass form:

\(\Sigma\vec{F}=m\vec{a}\)

Key ideas:

• net force causes acceleration
• acceleration is proportional to force
• acceleration is inversely proportional to mass
• force and acceleration are vectors

Newton’s Second Law is the central mathematical law of classical mechanics and forms the foundation for solving dynamics problems.