AP Physics 1 · Unit 2 · Topic 2.5

Newton’s Second Law

A nonzero net external force changes a system's velocity. For a constant-mass system, the acceleration of its center of mass points with the net force, grows in direct proportion to that force, and decreases as total mass increases.

Titration response · Newton's Second LawThe equivalence region produces the steepest change; its location depends on stoichiometry and acid-base strength.

Learning Goals

After this lesson, you should be able to:

apply Newton's second law to a clearly defined constant-mass system;
distinguish an individual force from the net external force;
write and solve independent component equations;
predict how acceleration changes when force or mass changes;
interpret force–acceleration and acceleration–mass graphs;
choose between single-object and multi-object system models; and
justify the direction of acceleration from a free-body diagram.

1. The Vector Law

For a constant-mass system observed from an inertial reference frame,

\[\sum\vec F_{\mathrm{ext}}=\vec F_{\mathrm{net}}=m_{\mathrm{sys}}\vec a_{\mathrm{cm}}\]

\[\vec a_{\mathrm{cm}}=\frac{\vec F_{\mathrm{net}}}{m_{\mathrm{sys}}}\]

The mass is a positive scalar, so acceleration and net force always point in the same direction. Individual parts may accelerate differently from the system's center of mass.

2. Net Force Is Not One More Force

The net force is the vector sum of all external interaction forces on the chosen system. It is not an additional arrow:

\[\vec F_{\mathrm{net}}=\vec F_1+\vec F_2+\vec F_3+\cdots\]

Internal forces between parts of a multi-object system are excluded from this external sum. Changing the system boundary can change which forces are internal, but the calculation must remain consistent with the chosen system.

3. Cause, Response, and Inertia

Quantity	Role in the model	Vector or scalar?
\(\vec F_{\mathrm{net}}\)	Net influence of the environment	Vector
\(\vec a_{\mathrm{cm}}\)	System's response: rate of velocity change	Vector
\(m_{\mathrm{sys}}\)	Measure of translational inertia	Positive scalar

The law predicts acceleration, not velocity. A leftward acceleration can describe an object moving right and slowing, moving left and speeding up, or momentarily at rest before moving left.

4. Force–Mass–Acceleration Comparison

**Newton's Second Law**Topic-matched fine pixel-art model · use the lesson's equations and labeled diagrams for exact scientific meaning.

5. A Reliable Solution Path

Choose the system and state its total mass.
Choose axes suited to the motion or constraints.
Draw a free-body diagram containing all external forces on that system.
Resolve angled forces into components without double-counting the original vector.
Write one second-law equation per axis.
Solve symbolically before inserting values when possible.
Check direction, units, and limiting behavior.

6. Component Form

The vector law becomes separate scalar equations:

\[\sum F_x=m a_x,\qquad \sum F_y=m a_y\]

Signs come from the coordinate system. An object can have \(a_y=0\) while \(a_x\ne0\), so vertical forces balance even though the full system is accelerating horizontally.

7. Worked Example: Competing Horizontal Forces

Question: A \(6.0\,\mathrm{kg}\) cart experiences \(34\,\mathrm N\) to the right and \(10\,\mathrm N\) to the left. Find its acceleration.

Choose right as positive:

\[\sum F_x=34-10=24\,\mathrm N\]

\[a_x=\frac{24}{6.0}=4.0\,\mathrm{m\,s^{-2}}\]

Answer: The cart accelerates at \(4.0\,\mathrm{m\,s^{-2}}\) to the right. Dividing either individual force by mass would not give the system's acceleration.

8. Worked Example: Two-Dimensional Net Force

Question: A \(5.0\,\mathrm{kg}\) puck experiences external forces \(\vec F_1=(15\hat i+20\hat j)\,\mathrm N\) and \(\vec F_2=(-5\hat i-5\hat j)\,\mathrm N\). Find its acceleration vector and magnitude.

\[\vec F_{\mathrm{net}}=(10\hat i+15\hat j)\,\mathrm N\]

\[\vec a=\frac{\vec F_{\mathrm{net}}}{m}=(2.0\hat i+3.0\hat j)\,\mathrm{m\,s^{-2}}\]

\[|\vec a|=\sqrt{2.0^2+3.0^2}=3.6\,\mathrm{m\,s^{-2}}\]

Answer: \(\vec a=(2.0\hat i+3.0\hat j)\,\mathrm{m\,s^{-2}}\), with magnitude approximately \(3.6\,\mathrm{m\,s^{-2}}\).

9. Proportional Reasoning Without Recalculating

From \(a=F_{\mathrm{net}}/m\), compare an original trial with a new trial:

\[\frac{a_2}{a_1}=\frac{F_{\mathrm{net},2}}{F_{\mathrm{net},1}}\frac{m_1}{m_2}\]

If the net force triples while mass doubles, then \(a_2/a_1=3/2\). This shortcut is valid only when the compared forces are net forces and the mass values refer to the complete chosen systems.

10. Graphical Models

\(a\) versus \(F_{\mathrm{net}}\) at fixed mass: a straight line through the origin with slope \(1/m\).
\(F_{\mathrm{net}}\) versus \(a\): a straight line through the origin with slope \(m\).
\(a\) versus \(1/m\) at fixed force: a straight line through the origin with slope \(F_{\mathrm{net}}\).
\(a\) versus \(m\) at fixed force: a decreasing inverse curve, not a straight line.

A nonzero graph intercept suggests an unaccounted force, sensor offset, or another systematic effect.

11. Worked Example: Two Carts as One System

Question: Carts of \(2.0\,\mathrm{kg}\) and \(3.0\,\mathrm{kg}\) remain in contact on a nearly frictionless track. A horizontal external force of \(20\,\mathrm N\) pushes the pair. Find the common acceleration.

Choose both carts as the system. Their mutual contact forces are internal, and total mass is \(5.0\,\mathrm{kg}\):

\[a=\frac{F_{\mathrm{ext}}}{m_1+m_2}=\frac{20}{2.0+3.0}=4.0\,\mathrm{m\,s^{-2}}\]

Answer: Both carts accelerate at \(4.0\,\mathrm{m\,s^{-2}}\). A separate single-cart equation would be needed to determine their contact force.

12. Net Force and Changing Velocity

Acceleration points with net force, but velocity need not. If \(\vec F_{\mathrm{net}}\) points opposite \(\vec v\), speed decreases. If it points with \(\vec v\), speed increases. If it is perpendicular to \(\vec v\), the velocity direction changes even if speed is momentarily unchanged.

Therefore “force in the direction of motion” is not a general rule; “acceleration in the direction of net force” is.

13. Investigation: Test the Functional Relationships

Use a cart, motion sensor, force sensor, and low-friction track.

Keep cart mass fixed and perform several trials with different approximately constant net forces.
Plot acceleration against measured net force and determine the best-fit slope.
Keep force approximately fixed, vary total cart mass, and plot acceleration against \(1/m\).
Compare the first slope with \(1/m\) and the second with \(F_{\mathrm{net}}\).
Use residuals and uncertainty to assess whether each linear model is supported.

Use the force measured during the analyzed interval rather than assuming a nominal pull is the net force.

14. Common Traps and Corrections

Trap: using one applied force as \(F_{\mathrm{net}}\). Correction: vector-sum every external force first.
Trap: adding net force as another free-body arrow. Correction: it is the resultant of existing arrows.
Trap: assuming acceleration points with velocity. Correction: acceleration points with net force.
Trap: using only one object's mass for a combined system. Correction: use the total mass inside the boundary.
Trap: mixing magnitudes and signed components. Correction: establish axes and preserve signs.
Trap: claiming \(a\) is directly proportional to mass. Correction: at fixed force, \(a\propto1/m\).

15. Key Takeaways

A system's center-of-mass acceleration is set by its net external force and total mass.
Net force and acceleration are parallel vectors.
Acceleration is directly proportional to net force and inversely proportional to mass.
Free-body diagrams supply the force terms in independent component equations.
A well-chosen multi-object system can eliminate internal forces from the calculation.
Force–acceleration data provide an experimental measurement of inertial mass.

Checkpoint

A system of mass \(m\) accelerates east at \(a\) under a net force \(F\). The net force is then reversed and doubled while the system mass becomes \(3m\). Determine the new acceleration vector in terms of \(a\), and justify both its magnitude and direction.

Concept scope cross-checked with the College Board AP Physics 1 course framework and the open educational references OpenStax University Physics: Newton's Second Law and OpenStax Physics: Newton's Second Law of Motion. Explanations, diagram, examples, and investigation are independently written for Insomnia Studyroom.

Official curriculum reference: College Board AP Physics 1 course page. The explanation and worked example are independently written for this study site.