Topic 2.5 Newton's Second Law: relate net force, mass and acceleration through the vector equation, apply it component by component, and extend it to connected systems and the general form with momentum.

What this topic is asking

The College Board (Topic 2.5) wants you to relate the net force, mass and acceleration through Newton's second law, to apply it component by component, and to extend it to connected systems. The course also expects the general form, that net force equals the time rate of change of momentum, which handles variable mass and variable forces. This is the central equation of dynamics, and nearly every problem in Units 2 to 4 uses it.

The second law as a vector equation

The equation is a vector statement, so it splits into one component equation per axis:

The standard procedure is to draw the free-body diagram, choose axes, resolve every force, sum the components on each axis, and set each sum equal to $m a$ on that axis. Often one axis has zero acceleration (giving a balance equation, for example for the normal force) while the other gives the acceleration. The first law is just the special case $\vec{a} = 0$.

The general form with momentum

AP Physics C also expects the more general statement of the law in terms of momentum $\vec{p} = m\vec{v}$:

When the mass is constant this becomes $\vec{F}_{net} = m\dfrac{d\vec{v}}{dt} = m\vec{a}$, the familiar form. The momentum form is essential when the force varies in time (you integrate it as an impulse) or when the mass changes (rockets, raindrops accreting mass), cases the course touches on. It also connects directly to the impulse-momentum theorem in Unit 4.

Connected systems

Many problems link objects with strings, pulleys or contact: Atwood machines, blocks on a table pulled by a hanging weight, stacked blocks. The efficient method has two steps. First, treat the whole system as a single mass to find the common acceleration: the internal tensions cancel, leaving the external forces over the total mass. Second, isolate one object and apply the second law to it alone to find the internal force (the tension or contact force). The two-step method avoids solving large simultaneous systems and is what AP graders expect.

Try this

Q1. A $4.0$ kg object experiences a net force of $20$ N east and $15$ N north simultaneously. Calculate the magnitude of its acceleration. [3 points]

• Cue. Net force $\sqrt{20^2 + 15^2} = 25$ N; $a = 25/4.0 = 6.25$ m/s squared.

Q2. State the general form of Newton's second law and explain when it differs from $\vec{F} = m\vec{a}$. [2 points]

• Cue. $\vec{F}_{net} = d\vec{p}/dt$; it differs from $m\vec{a}$ when the mass changes (for example a rocket losing fuel).