2.5 Newton’s Second Law

2.5 Newton’s Second Law

A velocity changes only when the net force—the vector sum of all external forces on a system—is nonzero. Newton’s Second Law turns that statement into a quantitative rule connecting force, mass, and acceleration.

Learning Objective 2.5.A: Describe the conditions under which a system’s velocity changes.

From unbalanced forces to acceleration

2.5.A.1: Unbalanced forces are a configuration of forces for which the net force on a system is not zero. Because force is a vector, forces pointing in opposite directions must be subtracted, while forces in the same direction add.

For a system with constant mass, Newton’s Second Law is

$$ \vec{F}_{\text{net}}=m\vec{a} $$

Equivalently,

$$ \vec{a}{\text{sys}}=\frac{\sum \vec{F}}{m{\text{sys}}} =\frac{\vec{F}{\text{net}}}{m{\text{sys}}}. $$

The acceleration points in the same direction as the net force. A larger net force produces a larger acceleration for the same mass; a larger mass produces a smaller acceleration for the same net force.

2.5.A.3: The velocity of a system’s center of mass changes only when a nonzero net external force acts on that system. Thus,

$$ \vec{F}_{\text{net}}=\vec{0} \quad\Longleftrightarrow\quad \vec{a}=\vec{0}, $$

provided the system’s mass is constant. Zero acceleration does not necessarily mean zero velocity: a cart moving at constant velocity has nonzero velocity but zero net force.

The momentum form—and the time-interval distinction

Momentum is defined as

$$ \vec{p}=m\vec{v}. $$

The change in momentum is the final momentum minus the initial momentum:

$$ \Delta\vec{p}=\vec{p}-\vec{p}_0. $$

The instantaneous, general form of Newton’s Second Law is

$$ \vec{F}_{\text{net}}=\frac{d\vec{p}}{dt}. $$

Over a finite time interval, however,

$$ \frac{\Delta\vec{p}}{\Delta t} $$

is the average net force during that interval, not necessarily the instantaneous force at every moment. If mass remains constant, then

$$ \vec{F}_{\text{net}} =m\frac{d\vec{v}}{dt} =m\vec{a}. $$

This distinction matters whenever force varies with time—for example, during a collision or while a motor gradually increases its thrust.

Reading acceleration from a velocity–time graph

Acceleration is the slope of a velocity-versus-time graph:

$$ a=\frac{\Delta v}{\Delta t} $$

for a straight segment, or the tangent slope at a particular instant on a curved graph. A horizontal segment has zero slope, so the acceleration and net force are zero at that time.

A graph that rises steadily represents constant positive acceleration and therefore a constant positive net force. A graph that slopes downward represents negative acceleration; the object may still be moving in the positive direction while slowing down.

Worked example: a block on a rough ramp

A block slides down a ramp of angle $\theta$ with kinetic-friction coefficient $\mu_k$. Choose down the ramp as positive. The forces parallel to the ramp are the downhill gravitational component $mg\sin\theta$ and the uphill friction force $f_k=\mu_k mg\cos\theta$.

The free-body representation gives

$$ \sum F_{\parallel}=mg\sin\theta-\mu_k mg\cos\theta. $$

Applying 2.5.A.2,

$$ ma=mg\sin\theta-\mu_k mg\cos\theta. $$

Divide by $m$:

$$ a=g\left(\sin\theta-\mu_k\cos\theta\right). $$

The mass cancels because both the gravitational and friction forces scale with mass. If $\theta=30^\circ$, $\mu_k=0.20$, and $g=9.8\ \text{m/s}^2$, then

$$ a=9.8\left(0.50-0.20(0.866)\right) \approx 3.2\ \text{m/s}^2. $$

If the block starts from rest and travels distance $L$, a kinematic relationship can then give its speed:

$$ v_A=\sqrt{2aL} =\sqrt{2gL\left(\sin\theta-\mu_k\cos\theta\right)}. $$

The derivation demonstrates 2.A: Derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway, rather than selecting a formula by appearance.

Proportional reasoning and evidence

Under 2.D: Predict new values or factors of change of physical quantities using functional dependence between variables, inspect

$$ a=\frac{F_{\text{net}}}{m}. $$

For fixed mass, doubling $F_{\text{net}}$ doubles $a$. For fixed net force, doubling $m$ halves $a$. Under 3.B: Apply an appropriate law, definition, theoretical relationship, or model to make a claim, Newton’s Second Law supports the claim that an object speeds up only when the net force has a component in its direction of motion.

Under 1.A: Create diagrams, tables, charts, or schematics to represent physical situations, represent the situation with a force diagram, a data table, or a graph of $a$ versus $F_{\text{net}}$. A Newton’s Second Law laboratory uses carts, tracks, and probeware to vary net force and mass, measure acceleration, and test the predicted relationships. Under 3.C: Justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws, the slope and pattern of the data become evidence for the model.

Misconception check

Misconception: “A force is required to keep an object moving.” A force is required to change velocity, not to maintain constant velocity. Another common error is treating $\Delta\vec{p}/\Delta t$ as the instantaneous law even when force varies; it is an interval average unless the force is constant.

Retrieval check: A cart’s velocity–time graph is horizontal at $t=4\ \text{s}$. What are its acceleration and net force at that instant? If its mass is tripled while the net force remains unchanged, what happens to its acceleration?

Answer: The graph’s zero slope means $a=0$, so $\vec{F}_{\text{net}}=\vec{0}$. Tripling the mass reduces the acceleration to one-third of its original value.