What Is Newton's Second Law of Motion, and Which Formula, Units, and Examples Describe It?
Newton's second law of motion states that the acceleration of an object is directly proportional to the net force on the object and inversely proportional to its mass. The formula is F_net = ma. The momentum form is F_net = dp/dt. The SI unit of force is the newton (N), equal to 1 kg·m/s². Acceleration points in the direction of the net force. Isaac Newton published the law in 1687 in the Principia.
For a reliable reference, consult OpenStax section on forces and Newton’s laws.
What Is Newton's Second Law of Motion?
Newton's second law of motion states that the net force on an object equals the object's mass multiplied by its acceleration. The law links the cause of a velocity change (force) to the effect (acceleration).
According to OpenStax University Physics Volume 1, the acceleration of a system is directly proportional to the net external force and inversely proportional to the mass of the system. Isaac Newton published the law in 1687 in Philosophiæ Naturalis Principia Mathematica. The original statement ties the change of motion to the applied force and places the change along the line of that force. The law is the second of Newton's 3 laws of motion.
What Is the Formula for Newton's Second Law?
The formula for Newton's second law is F_net = ma, where F_net is net force, m is mass, and a is acceleration. The momentum form is F_net = dp/dt.
The table lists 7 forms of the law and the situation that each form fits.
| Form | Quantities | Use Case |
|---|---|---|
| F_net = ma | Net force (N), mass (kg), acceleration (m/s²) | Constant mass |
| a = F_net/m | Acceleration (m/s²) | Acceleration from force and mass |
| m = F_net/a | Mass (kg) | Mass from force and acceleration |
| ΣFₓ = maₓ and ΣF_y = ma_y | Force and acceleration components (N, m/s²) | Forces at angles |
| F_net = m dv/dt | Velocity v (m/s), time t (s) | Calculus form for constant mass |
| F_net = Δp/Δt | Momentum change Δp (kg·m/s), time interval Δt (s) | Average force during a collision |
| F_net = dp/dt | Momentum p (kg·m/s) | General form, including variable mass |
What Do the Symbols in Newton's Second Law Mean?
The table defines each quantity in F_net = ma and lists its SI unit.
| Quantity | Symbol | SI Unit | Definition |
|---|---|---|---|
| Net force | F_net or ΣF | newton (N) | Vector sum of all forces on the object |
| Mass | m | kilogram (kg) | Amount of matter and measure of inertia |
| Acceleration | a | meter per second squared (m/s²) | Rate of change of velocity |
| Momentum | p | kilogram meter per second (kg·m/s) | Product of mass and velocity |
| Time | t | second (s) | Elapsed time |
Is Newton's Second Law F = ma or F = dp/dt?
Newton's second law is F_net = dp/dt in its general form, and F_net = ma applies when mass stays constant. Differentiating p = mv gives dp/dt = m(dv/dt) + v(dm/dt). The second term equals zero when mass is constant, which leaves F_net = ma.
Variable-mass systems use the momentum form on a defined system. Examples include rockets that expel fuel and conveyor belts that receive falling material.
What Is the SI Unit of Force in Newton's Second Law?
The SI unit of force is the newton (N), defined as 1 kg·m/s². One newton is the net force that gives a 1 kg mass an acceleration of 1 m/s².
The table converts 4 force units to newtons.
| Unit | Symbol | Value in Newtons |
|---|---|---|
| Newton | N | 1 N |
| Dyne | dyn | 0.00001 N |
| Pound-force | lbf | 4.448 N |
| Kilogram-force | kgf | 9.80665 N |
In Which Direction Does Acceleration Point in Newton's Second Law?
Acceleration points in the direction of the net force. Force and acceleration are vectors, so the law holds along each axis separately.
An object that moves east and receives a net force toward the west slows down, with acceleration pointing west. A net force perpendicular to velocity changes the direction of motion and leaves the speed unchanged.
How Do Force, Mass, and Acceleration Relate in Newton's Second Law?
Acceleration is directly proportional to net force at constant mass and inversely proportional to mass at constant net force.
The table lists the acceleration for 6 combinations of net force and mass.
| Net Force (N) | Mass (kg) | Acceleration (m/s²) | Change Compared with Row 2 |
|---|---|---|---|
| 0 | 2 | 0 | Not applicable |
| 4 | 2 | 2 | Baseline |
| 8 | 2 | 4 | Force doubled, acceleration doubled |
| 12 | 2 | 6 | Force tripled, acceleration tripled |
| 4 | 4 | 1 | Mass doubled, acceleration halved |
| 4 | 8 | 0.5 | Mass quadrupled, acceleration quartered |
A graph of acceleration versus net force at constant mass is a straight line through the origin with slope 1/m. A graph of net force versus acceleration has slope m, which gives a way to measure mass.
How Is Newton's Second Law Derived?
Newton's second law follows from 2 experimental proportionalities and a unit definition, in 5 steps.
- Measure acceleration at constant mass for several forces, which gives a ∝ F.
- Measure acceleration at constant force for several masses, which gives a ∝ 1/m.
- Combine the proportionalities into a ∝ F/m.
- Define the newton so that the proportionality constant equals 1.
- Rearrange to obtain F_net = ma.
A dynamics cart experiment shows the first proportionality. A 1.0 kg system pulled by forces of 1.0 N, 2.0 N, 3.0 N, and 4.0 N has ideal accelerations of 1.0, 2.0, 3.0, and 4.0 m/s². Friction in real tracks lowers the measured values.
How to Apply Newton's Second Law to a Problem?
Applying Newton's second law takes 5 steps: identify, draw, choose, write, and solve.
- Identify the object and every force acting on it.
- Draw a free-body diagram with one arrow per force.
- Choose a positive direction for each axis.
- Write ΣFₓ = maₓ and ΣF_y = ma_y.
- Solve the equations for the unknown quantity.
A free-body diagram is a sketch that represents the object as a dot and each force as an arrow. Only forces that act on the object appear in the diagram.
What Are Examples of Newton's Second Law With Calculations?
Examples of Newton's second law include force from acceleration, acceleration from force, mass from force, friction problems, inclines, elevators, and connected masses. The 12 worked examples use g = 9.81 m/s².
How to Find Force With Newton's Second Law?
Worked example 1: a 1,200 kg car accelerates at 2.5 m/s². F_net = 1,200 × 2.5 = 3,000 N in the direction of the acceleration.
How to Find Acceleration With Newton's Second Law?
Worked example 2: a net force of 15 N acts on a 3.0 kg object. a = 15/3.0 = 5.0 m/s² in the direction of the force.
How to Find Mass With Newton's Second Law?
Worked example 3: a net force of 20 N produces an acceleration of 4.0 m/s². m = 20/4.0 = 5.0 kg.
How to Apply Newton's Second Law With Friction?
Worked example 4: a 40 kg crate receives a 150 N push, and friction exerts 50 N in the opposite direction.
- Net force: 150 N − 50 N = 100 N forward
- Acceleration: a = 100/40 = 2.5 m/s²
How to Apply Newton's Second Law in Two Dimensions?
Worked example 5: a 2.0 kg object receives a 6.0 N force east and an 8.0 N force north.
- Net force magnitude: √(6.0² + 8.0²) = 10.0 N
- Acceleration magnitude: a = 10.0/2.0 = 5.0 m/s²
- Direction: tan⁻¹(8.0/6.0) = 53.1 degrees north of east
How Does Newton's Second Law Explain Free Fall?
Worked example 6: the net force on a falling object without air resistance is its weight, mg. a = mg/m = g = 9.81 m/s². Acceleration does not depend on mass, so a 1 kg object and a 10 kg object fall with the same acceleration in a vacuum.
How to Apply Newton's Second Law on a Frictionless Incline?
Worked example 7: a block rests on a frictionless 30 degree incline. The net force along the incline is mg sin 30°, so a = g sin 30° = 9.81 × 0.500 = 4.9 m/s² down the incline.
How to Apply Newton's Second Law in an Elevator?
Worked example 8: a 70 kg person rides an elevator that accelerates downward at 1.5 m/s².
- Weight: 70 × 9.81 = 686.7 N down
- Equation: mg − N = ma
- Normal force: N = m(g − a) = 70 × (9.81 − 1.5) = 581.7 N up
The scale reading is 581.7 N, which is lower than the 686.7 N weight.
How to Apply Newton's Second Law to an Atwood Machine?
Worked example 9: masses of 4.0 kg and 2.0 kg hang from a light rope over a frictionless pulley.
- Acceleration: a = (m₁ − m₂)g/(m₁ + m₂) = (2.0 × 9.81)/6.0 = 3.27 m/s²
- Tension: T = m₂(g + a) = 2.0 × (9.81 + 3.27) = 26.2 N
How to Apply Newton's Second Law to Two Touching Blocks?
Worked example 10: a 10 N force pushes a 2.0 kg block that touches a 3.0 kg block on a frictionless surface.
- System acceleration: a = 10/(2.0 + 3.0) = 2.0 m/s²
- Contact force on the 3.0 kg block: 3.0 × 2.0 = 6.0 N
How to Apply the Momentum Form of Newton's Second Law?
Worked example 11: a 0.057 kg tennis ball leaves a racket at 50 m/s from rest, with a contact time of 0.005 s.
- Momentum change: Δp = 0.057 × 50 = 2.85 kg·m/s
- Average force: F = 2.85/0.005 = 570 N
How to Apply Newton's Second Law With Air Resistance?
Worked example 12: an 80 kg skydiver experiences 500 N of upward air resistance.
- Weight: 80 × 9.81 = 784.8 N down
- Net force: 784.8 − 500 = 284.8 N down
- Acceleration: a = 284.8/80 = 3.56 m/s² down
Air resistance increases with speed. The net force reaches 0 N at terminal velocity, where the skydiver falls at constant speed.
What Is the Rotational Form of Newton's Second Law?
The rotational form of Newton's second law is Στ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration.
The table maps each linear quantity to its rotational counterpart.
| Linear Quantity | Symbol and Unit | Rotational Quantity | Symbol and Unit |
|---|---|---|---|
| Net force | F_net (N) | Net torque | τ_net (N·m) |
| Mass | m (kg) | Moment of inertia | I (kg·m²) |
| Acceleration | a (m/s²) | Angular acceleration | α (rad/s²) |
| Momentum | p (kg·m/s) | Angular momentum | L (kg·m²/s) |
Worked example: a solid disk of mass 2.0 kg and radius 0.30 m has I = ½MR² = 0.090 kg·m². A net torque of 0.45 N·m gives α = 0.45/0.090 = 5.0 rad/s².
How Does Newton's Second Law Relate to the First and Third Laws?
Newton's first law is the special case of the second law where the net force equals zero, and the third law describes force pairs on different objects.
The table compares the 3 laws across 4 attributes.
| Attribute | First Law | Second Law | Third Law |
|---|---|---|---|
| Statement | Velocity stays constant when net force is 0 N | F_net = ma | Forces come in equal and opposite pairs |
| Net force | 0 N | Any value | Not applicable |
| Key concept | Inertia | Acceleration | Interaction |
| Example | Book at rest on a table | Rocket at liftoff | Swimmer pushing off a wall |
The 2 forces of a third-law pair act on different objects, so the pair does not cancel in the net force on one object.
What Is the Difference Between Mass and Weight in Newton's Second Law?
Mass is the amount of matter in kilograms, and weight is the gravitational force in newtons, W = mg. Weight is a case of Newton's second law with a = g.
The table lists the weight of a 10 kg object at 3 locations.
| Location | Gravitational Acceleration g (m/s²) | Weight of 10 kg (N) |
|---|---|---|
| Earth surface | 9.81 | 98.1 |
| Moon surface | 1.62 | 16.2 |
| Mars surface | 3.71 | 37.1 |
Mass stays at 10 kg at every location. Weight changes with g.
When Does Newton's Second Law Not Apply?
Newton's second law in the form F = ma holds in inertial frames, at speeds far below the speed of light, and for objects much larger than atoms.
The table lists 4 conditions and the replacement framework.
| Condition | Effect on F = ma | Replacement or Adjustment |
|---|---|---|
| Non-inertial frame (rotating or accelerating) | Fictitious forces appear | Add fictitious forces, such as the centrifugal force |
| Speeds near the speed of light | Mass-velocity relation changes | Use F = dp/dt with p = γmv in special relativity |
| Atomic and subatomic scales | Particle behavior is probabilistic | Use quantum mechanics |
| Variable mass | F = ma does not hold for the changing system | Use F = dp/dt on a defined system |
The Lorentz factor γ equals 1.005 at 10% of the speed of light, so the relativistic correction is about 0.5% at that speed.
What Are the Applications of Newton's Second Law?
Newton's second law applies in 5 fields: vehicle safety, aerospace, sports, structural engineering, and robotics.
- Vehicle safety: a 1,200 kg car at 25 m/s that brakes with 6,000 N has a = 5.0 m/s² and a stopping distance of v²/(2a) = 62.5 m. Airbags extend the collision time, which lowers the average force in F = Δp/Δt.
- Aerospace: rocket thrust minus weight and drag gives the net force that sets the acceleration.
- Sports: a longer follow-through increases the contact time, which changes the force a ball receives for a given momentum change.
- Structural engineering: static structures satisfy ΣF = 0 in each direction.
- Robotics: motor torque and arm inertia determine angular acceleration through Στ = Iα.
Who Formulated Newton's Second Law, and When?
Isaac Newton published the second law in 1687 in the Principia. The statement relates the change of motion to the applied force. Leonhard Euler wrote the equation in its modern differential form for a point mass in 1750.
Galileo Galilei described uniform acceleration in free fall in the early 1600s. His work on falling bodies preceded Newton's law.
What Are Common Misconceptions About Newton's Second Law?
Common misconceptions about Newton's second law involve force and motion, falling objects, and the meaning of F.
The table lists 6 misconceptions and the corrected statement for each.
| Misconception | Correct Statement |
|---|---|
| A force is required to keep an object moving | Constant velocity requires a net force of 0 N |
| Heavier objects fall faster in a vacuum | The net force mg and the mass m give a = g for every mass |
| F in F = ma is any single force | F is the net force, the vector sum of all forces |
| Mass and weight are the same | Mass is in kilograms, and weight is a force in newtons |
| Acceleration follows the direction of velocity | Acceleration follows the direction of the net force |
| A large force always produces a large acceleration | Acceleration equals force divided by mass, so a large mass lowers it |
People Also Ask About Newton's Second Law of Motion
What is Newton's second law of motion in simple terms?
Newton's second law states that a larger net force gives a larger acceleration, and a larger mass gives a smaller acceleration.
What is the formula for Newton's second law?
The formula is F_net = ma.
What is the unit of force in Newton's second law?
The unit of force is the newton (N), equal to 1 kg·m/s².
What is the momentum form of Newton's second law?
The momentum form is F_net = dp/dt, where p is momentum.
Is Newton's second law a vector equation?
Yes, force and acceleration are vectors, so the law holds along each axis.
What is the direction of acceleration in Newton's second law?
Acceleration points in the direction of the net force.
What happens to acceleration when mass doubles?
Acceleration halves at constant net force.
What happens to acceleration when force doubles?
Acceleration doubles at constant mass.
How does Newton's second law explain free fall?
The net force is mg, so a = g = 9.81 m/s² for every mass without air resistance.
What is the rotational form of Newton's second law?
The rotational form is Στ = Iα.
Does Newton's second law apply at the speed of light?
No, special relativity replaces F = ma with F = dp/dt and p = γmv at speeds near the speed of light.
What is the difference between Newton's first and second laws?
The first law describes motion with zero net force, and the second law relates a nonzero net force to acceleration.
What is Newton's second law of motion in one sentence?
Newton's second law of motion states that the net force on an object equals its mass times its acceleration.
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