Newton’s Second Law Connects Force to Motion

Motion does not require a force. Changing motion does. Newton’s second law identifies the net external force as the cause of changing momentum, turning a physical interaction into a predictive equation.

The general statement

Momentum is $\mathbf p=m\mathbf v$. Newton’s second law states

When mass is constant, this becomes

$$\mathbf F_{\mathrm{net}}=m\frac{d\mathbf v}{dt}=m\mathbf a.$$

The familiar $\mathbf F=m\mathbf a$ is therefore an important special case of the momentum law.

Net force, not individual force

Suppose a $10\,\mathrm{kg}$ crate is pushed right with $50\,\mathrm N$ while friction pushes left with $20\,\mathrm N$. The net force is $30\,\mathrm N$ right, so

$$a=\frac{F_{\mathrm{net}}}{m}=3\,\mathrm{m/s^2}.$$

The acceleration points with the net force, regardless of the crate’s current direction of motion.

A recipe for force problems

First choose the object whose motion you are studying. Draw only forces exerted on that object. Choose coordinate directions, resolve each force into components, add the components, and apply the law separately along each axis.

For equilibrium, $\mathbf a=0$, so $\mathbf F_{\mathrm{net}}=0$. Zero net force permits both rest and constant-velocity motion.

From force to a trajectory

Since acceleration is the second derivative of position,

$$m\frac{d^2\mathbf x}{dt^2}=\mathbf F_{\mathrm{net}},$$

Newton’s law is a differential equation. Given the forces and initial motion, solving it predicts the object’s path.

Let the free-body diagram lead

Choose one system. Every arrow must name an external agent. Do not draw “$ma$” as a force; acceleration follows from the vector sum.

Check your understanding

Are a resting book’s weight and normal force a third-law pair?

Show the reasoning

No. Both act on the book. Third-law partners act on different objects.