4 -Newton’s Laws of Motion

A structured guide to using Newton’s laws, free-body diagrams, force models, and component equations to analyze motion in AP Physics 1.

Forces, Systems, and Equilibrium

Newton’s laws provide a model for predicting how external interactions change motion. The central strategy is to identify the system, represent each external interaction as a force, and use components of .

A force is a vector interaction, so it has both magnitude and direction. The SI unit of force is the newton:

1\ \text{N}=1\ \text{kg}\cdot\text{m/s}^2

The is the vector sum of all external forces:

\vec F_{\text{net}}=\sum\vec F

Only the determines acceleration. Several forces may act on an object while canceling, or they may combine to produce a in any direction.

Choose the system

The system is the object or collection of objects being analyzed. Forces exerted by objects outside the system are external forces and belong in the force analysis. Forces between parts of the system are internal; when the entire collection is treated as one system, internal forces cancel in pairs and are not included in the system’s overall Newton’s second-law equation.

First law and equilibrium

Newton’s first law states that an object remains at rest or moves with constant velocity when the net external force is zero:

\vec F_{\text{net}}=0\quad\Longrightarrow\quad \vec a=0

An object with zero acceleration may be stationary or moving at constant velocity. Equilibrium does not mean that no forces act; it means that the forces cancel as vectors.

Takeaway: Start every problem by defining the system and determining the vector sum of its external forces.

Free-Body Diagrams and Components

A isolates one selected object and displays every external force acting on it. It is not a drawing of the object’s path or motion.

Construction procedure

Choose the object or system.
Select coordinate axes, preferably aligning an axis with the motion, surface, or another convenient direction.
Identify all physical interactions, including contact interactions and long-range forces.
Draw one force vector for each interaction and point it in the physically correct direction.
Resolve angled forces into components when needed.
Write a separate Newton’s second-law equation for each coordinate direction.

For a box pulled across a rough horizontal floor, the forces may include an applied force, , a , and weight. The component equations are

\sum F_x=F_{\text{app}}-f=ma_x

\sum F_y=N-mg=ma_y

If the box remains on the horizontal surface, $a_y=0$ , so $N=mg$ in that particular situation. The is not always equal to the weight; additional vertical forces or vertical acceleration can change it.

Do not draw acceleration as a force. Acceleration is the result of the , not an additional interaction on the .

Takeaway: A complete diagram comes from physical interactions, not from the object’s motion alone.

Newton’s Laws and Force Pairs

connects force, mass, and acceleration:

\boxed{\vec F_{\text{net}}=m\vec a}

In component form,

\sum F_x=ma_x,\qquad \sum F_y=ma_y

The acceleration points in the direction of the . The velocity and acceleration do not have to point in the same direction, so an object can slow down even while moving in a particular direction.

Newton’s third-law pairs

concerns pairs of forces from one interaction:

\vec F_{A\to B}=-\vec F_{B\to A}

The two forces have equal magnitudes, opposite directions, occur simultaneously, and act on different objects. Therefore, they do not cancel on one .

For a book resting on a table, the table’s upward force on the book is paired with the book’s downward force on the table. The book’s weight is Earth’s gravitational force on the book, so its third-law partner is the book’s gravitational force on Earth. These are two different interactions.

Common conceptual distinction

Mass is an intrinsic property measured in kilograms. Weight is a gravitational force measured in newtons. Near Earth’s surface,

\vec w=m\vec g

where $g\approx9.8\ \text{m/s}^2$ downward, and the magnitude is $w=mg$ .

Takeaway: Apply to one selected system, and identify third-law partners by checking that the forces act on different objects.

Common Forces and Friction

The force model must match the physical interaction. Common forces include the following.

Weight: The gravitational force exerted by a nearby astronomical body. Near Earth, its magnitude is $w=mg$ and its direction is downward.
: A contact force perpendicular to a surface. For an object at rest on a horizontal surface with no other vertical forces, $N=mg$ , but this equality is not universal.
Tension: A pulling force transmitted along a stretched rope, cable, or string. For an ideal massless rope over a frictionless pulley, the tension has the same magnitude throughout the rope.
Spring force: An ideal restoring force described by Hooke’s law,
$\vec F_s=-k\vec x$
where $k$ is the spring constant and $\vec x$ is displacement from the relaxed position.
Applied force: A direct push or pull exerted by a person or another object.
Thrust: A propulsive force, such as the force exerted by expelled gas on a rocket.
Drag: A resistive force caused by motion through a fluid. It generally acts opposite the object’s velocity and may be neglected when a problem says to ignore air resistance.

Friction

Friction acts parallel to contacting surfaces and opposes relative slipping or the tendency toward relative slipping. It is not always opposite the object’s velocity.

For surfaces that do not slide, adjusts within the range

0\le f_s\le f_{s,\max}

with

f_{s,\max}=\mu_sN

For surfaces that are sliding, is modeled as

f_k=\mu_kN

Usually, $\mu_k<\mu_s$ . Do not set equal to $\mu_sN$ unless the object is at the threshold of slipping.

Takeaway: Identify the source, direction, and contact conditions of each force before assigning an equation.

Newtonian Applications

Many applications become straightforward when the axes and constraints are chosen carefully.

Inclined planes

For a block on a frictionless incline, choose one axis parallel and one perpendicular to the surface. Resolve the weight into components:

w_{\parallel}=mg\sin\theta

w_{\perp}=mg\cos\theta

Along the incline,

mg\sin\theta=ma_{\parallel}

a_{\parallel}=g\sin\theta

Perpendicular to the incline, if there is no acceleration away from or into the surface,

N-mg\cos\theta=0

and therefore $N=mg\cos\theta$ . If the block slides down with , use

mg\sin\theta-f_k=ma_{\parallel}

Connected objects

Draw a separate diagram and equation for each object. For a mass $m_1$ on a frictionless table connected to a hanging mass $m_2$ , the equations can be written as

T=m_1a

m_2g-T=m_2a

Adding them eliminates the internal tension:

m_2g=(m_1+m_2)a

Thus,

a=\frac{m_2g}{m_1+m_2}

The objects can share an acceleration because of the rope constraint, even though their force equations differ.

Elevators and apparent weight

For a person of mass $m$ in an elevator, taking upward as positive,

N-mg=ma_y

N=m(g+a_y)

The person feels heavier when the elevator accelerates upward and lighter when it accelerates downward. At constant velocity, $a_y=0$ and $N=mg$ .

Circular motion

Circular motion requires inward acceleration:

a_c=\frac{v^2}{r}

The phrase centripetal force describes the required net inward force, not a new type of force:

\sum F_{\text{inward}}=m\frac{v^2}{r}

Gravity, tension, friction, or a may provide this inward depending on the situation.

Takeaway: Choose coordinates that simplify the geometry, then apply along each relevant direction.

Problem-Solving Workflow

Use the following workflow to organize nearly any force problem.

Describe the motion and choose the system.
Draw a complete containing only external forces on that system.
Choose positive axes and state the sign convention.
Resolve angled forces into components.
Write $\sum F_x=ma_x$ and $\sum F_y=ma_y$ .
Add constraints such as equilibrium, a common acceleration, an inextensible rope, or an appropriate friction model.
Solve algebraically before substituting numerical values when practical.
Check units, signs, limiting cases, and physical reasonableness.

Frequent errors to avoid

Treating mass and weight as the same quantity.
Assuming $N=mg$ in every situation.
Drawing acceleration as a force.
Pairing two forces on the same object as a Newton’s third-law pair.
Using $f_s=\mu_sN$ when has not reached its maximum value.
Assuming friction is always opposite the object’s velocity rather than opposite relative slipping.
Calling centripetal force a separate force instead of identifying the actual inward .

A strong final check asks whether the direction of the calculated acceleration agrees with the and whether special cases behave sensibly. For example, if the is zero, the result should give $a=0$ , even if several individual forces are present.

Final takeaway: Accurate system selection, complete force diagrams, correct components, and careful sign conventions turn Newton’s laws into a reliable problem-solving method.