2.2 Second law of motion

Newton's second law of motion quantifies how forces change an object's motion. It connects force, mass, and acceleration through one equation, and nearly every problem you'll solve in mechanics depends on it.

Definition of Newton's second law

Newton's second law says that the net force on an object equals its mass times its acceleration: $\vec{F}_{net} = m\vec{a}$. This is the quantitative link between "how hard you push" and "how much the motion changes."

Force and acceleration relationship

The acceleration of an object points in the same direction as the net force and is directly proportional to it. If you double the net force on an object (keeping mass constant), the acceleration doubles too.

• This means a 10 N net force on a 2 kg block produces $a = \frac{10}{2} = 5 \text{ m/s}^2$. Push with 20 N instead, and you get 10 m/s².
• Direction matters: force and acceleration are both vectors, so a force to the right produces acceleration to the right.

Mass as inertial property

Mass measures how much an object resists acceleration. A heavier object needs more force to achieve the same acceleration as a lighter one.

• A 1 kg ball and a 10 kg ball both pushed with 10 N will accelerate at 10 m/s² and 1 m/s², respectively.
• In this course, mass stays constant (we're not dealing with speeds near the speed of light).

Mathematical formulation

The math behind the second law gives you a precise tool for predicting motion whenever you know the forces involved.

Vector equation F = ma

The full vector form is:

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

Here $\vec{F}_{net}$ is the vector sum of all forces on the object, $m$ is its mass, and $\vec{a}$ is the resulting acceleration vector. Because it's a vector equation, it captures both magnitude and direction at once.

Scalar equations for components

In practice, you break the vector equation into components along your chosen axes:

$F_x = ma_x \quad,\quad F_y = ma_y \quad,\quad F_z = ma_z$

This is where the real problem-solving happens. For a block on a flat surface, you might only need the x and y equations. For a block on a ramp, you'd pick axes parallel and perpendicular to the slope so that the math stays clean.

Applications of second law

Free-body diagrams

A free-body diagram (FBD) is a sketch showing every force acting on a single object. Drawing one is the first step in almost every Newton's second law problem.

1. Isolate the object of interest (draw it as a dot or simple shape).
2. Identify every force: weight ($mg$, downward), normal force (perpendicular to surface), friction (along surface, opposing motion or tendency to move), tension, applied pushes/pulls, etc.
3. Draw each force as an arrow starting from the object, with length roughly proportional to magnitude.
4. Label every arrow with a symbol ($\vec{W}$, $\vec{N}$, $\vec{f}$, $\vec{T}$, etc.).

If a force is missing from your FBD, it won't appear in your equations, and your answer will be wrong. Take FBDs seriously.

Multiple forces on objects

Most real problems involve more than one force. To find the acceleration, you add all force vectors to get the net force, then apply $\vec{F}_{net} = m\vec{a}$.

For example, consider a 5 kg box on a table being pushed with 30 N to the right while friction pushes back with 10 N. The net force is $30 - 10 = 20$ N to the right, so $a = \frac{20}{5} = 4 \text{ m/s}^2$ to the right.

For more complex setups (pulley systems, objects connected by ropes), you draw a separate FBD for each object and write a separate $F = ma$ equation for each. Then solve the system of equations together.

Impulse and momentum

The second law can also be expressed in terms of momentum, which is especially useful when forces act over short time intervals (like collisions).

Impulse-momentum theorem

Momentum is defined as $\vec{p} = m\vec{v}$. Newton's second law in its more general form says that net force equals the rate of change of momentum:

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

Integrating both sides over a time interval gives the impulse-momentum theorem:

$\vec{J} = \int \vec{F} \, dt = \Delta \vec{p} = m\Delta \vec{v}$

The impulse $\vec{J}$ (force multiplied by the time it acts) equals the change in momentum. This explains why airbags work: they increase the collision time, reducing the average force on you for the same change in momentum.

Conservation of momentum

When no net external force acts on a system, the total momentum stays constant:

$\vec{p}_{total,\,before} = \vec{p}_{total,\,after}$

For a two-object collision: $m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}_1' + m_2\vec{v}_2'$

This principle comes directly from Newton's second and third laws. Internal forces between objects in the system always come in equal-and-opposite pairs (third law), so they cancel out and can't change the system's total momentum.

Friction and second law

Friction is a contact force that resists sliding between surfaces. Almost every real-world second law problem involves friction, so you need to be comfortable including it.

Static vs kinetic friction

• Static friction ($f_s$) keeps an object from starting to slide. It adjusts to match the applied force up to a maximum: $f_s \leq \mu_s N$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force.
• Kinetic friction ($f_k$) acts on an object that's already sliding: $f_k = \mu_k N$. It has a fixed magnitude (for a given pair of surfaces) and always opposes the direction of motion.
• Typically $\mu_s > \mu_k$, which is why it's harder to start pushing a heavy box than to keep it moving.

Friction on inclined planes

On a ramp tilted at angle $\theta$:

1. Gravity pulls the object straight down with force $mg$.
2. Break gravity into components: $mg\sin\theta$ parallel to the ramp (pulling the object down the slope) and $mg\cos\theta$ perpendicular to the ramp.
3. The normal force equals $N = mg\cos\theta$ (assuming no acceleration perpendicular to the surface).
4. Friction acts along the surface: $f = \mu N = \mu mg\cos\theta$.
5. Apply Newton's second law along the ramp: $mg\sin\theta - f = ma$ (for an object sliding down).

The angle of repose is the steepest angle at which an object can sit on a ramp without sliding. At that angle, $\tan\theta = \mu_s$.

Circular motion

When an object moves in a circle at constant speed, it's still accelerating because its direction keeps changing. Newton's second law tells us there must be a net force causing that acceleration.

Centripetal force

The acceleration for uniform circular motion points toward the center of the circle and has magnitude $a_c = \frac{v^2}{r}$. Applying the second law:

$F_c = \frac{mv^2}{r}$

"Centripetal force" isn't a new type of force. It's just the name for whatever real force (or combination of forces) points inward: tension in a string, gravity for an orbiting satellite, friction for a car turning on a road.

Banked curves and friction

On a banked (tilted) curve, the normal force has a horizontal component that points toward the center of the turn. This provides some or all of the centripetal force, reducing how much friction is needed.

At the "ideal" banking angle, no friction is required at all. That angle satisfies $\tan\theta = \frac{v^2}{rg}$. Highway on-ramps and racetracks are banked for this reason.

Systems of particles

Newton's second law extends naturally to systems of multiple objects. Instead of tracking every particle individually, you can often analyze the system as a whole.

Center of mass

The center of mass is the average position of all the mass in a system. For the system as a whole, Newton's second law becomes:

$\vec{F}_{ext,\,net} = M\vec{a}_{cm}$

where $M$ is the total mass and $\vec{a}_{cm}$ is the acceleration of the center of mass. The center of mass moves as if all the mass were concentrated there and all external forces acted on that point.

Internal vs external forces

• Internal forces act between objects within the system (e.g., tension in a rope connecting two blocks). By Newton's third law, these always come in equal-and-opposite pairs and cancel when you sum forces for the whole system.
• External forces come from outside the system (gravity, a push from your hand, friction with the ground). Only external forces can change the total momentum or accelerate the center of mass.

This distinction is why you can analyze a system of connected blocks as one object when you only care about the system's overall acceleration.

Variable mass systems

Some problems involve objects whose mass changes over time. The standard $F = ma$ needs to be adapted because $m$ is no longer constant.

Rocket propulsion

A rocket accelerates by expelling mass (exhaust gas) at high speed. The thrust equation is:

$F_{thrust} = v_e \frac{dm}{dt}$

where $v_e$ is the exhaust velocity relative to the rocket and $\frac{dm}{dt}$ is the rate of mass ejection. The rocket doesn't need anything to "push against." It works purely through conservation of momentum: mass goes backward, rocket goes forward.

Conveyor belts

Conveyor belts are another variable-mass scenario. As material is loaded onto a moving belt, the belt must exert extra force to accelerate the new material up to belt speed. The analysis uses the same principle: account for both the acceleration of existing mass and the momentum carried by mass entering or leaving the system.

Limitations and extensions

Newton's second law works extremely well for everyday speeds and sizes, but it has boundaries.

Relativistic effects

At speeds approaching the speed of light ($c \approx 3 \times 10^8$ m/s), the relationship between force and acceleration changes. Momentum becomes $\vec{p} = \gamma m\vec{v}$, where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$. For this intro course, you won't need relativistic corrections, but it's worth knowing that $F = ma$ is an approximation that works perfectly at normal speeds.

Quantum mechanical considerations

At atomic and subatomic scales, classical mechanics breaks down entirely. Particles don't have definite positions and velocities simultaneously (Heisenberg's uncertainty principle), so the concept of a deterministic force producing a deterministic acceleration no longer applies. Again, this is well outside the scope of this course, but it sets the stage for why physics has multiple frameworks.

Problem-solving strategies

A consistent approach to second law problems will save you time and reduce errors.

Step-by-step method

1. Read the problem carefully. Identify what you're solving for and what information is given.
2. Draw a free-body diagram for each object of interest. Label every force.
3. Choose a coordinate system. Align one axis with the direction of acceleration when possible (e.g., along a ramp for incline problems, toward the center for circular motion).
4. Write $F = ma$ for each axis and for each object. Substitute known force expressions ($mg$, $\mu N$, $\frac{mv^2}{r}$, etc.).
5. Solve the equations. If you have multiple unknowns, you'll need multiple equations (one per object or per axis).
6. Check your answer. Do the units work out? Is the direction reasonable? Does the magnitude make physical sense?

Choosing appropriate coordinate systems

Your choice of axes can make a problem easy or painful. A few guidelines:

• For flat surfaces, use horizontal and vertical axes.
• For inclined planes, use axes parallel and perpendicular to the slope. This way gravity is the only force you need to decompose, and the acceleration lies along one axis.
• For circular motion, use radial (toward center) and tangential axes.
• If the problem has symmetry, exploit it. The right coordinate system can eliminate an entire equation.

Historical context

Newton's contributions

Isaac Newton published his three laws of motion and the law of universal gravitation in Principia Mathematica (1687). He also co-developed calculus, which provided the mathematical language needed to describe changing motion. His framework unified terrestrial physics (falling apples) and celestial mechanics (orbiting planets) under the same set of laws.

Development of classical mechanics

Newton built on the work of Galileo (who studied acceleration and inertia), Kepler (who described planetary orbits), and Descartes (who contributed to the concept of momentum). Later physicists like Euler, Lagrange, and Hamilton reformulated and extended classical mechanics into more powerful mathematical forms. Classical mechanics remained the dominant framework in physics until the early 20th century, when relativity and quantum mechanics revealed its limits at extreme speeds and tiny scales. For the vast majority of engineering and everyday applications, though, Newton's laws remain the go-to tools.