Newton's laws of motion

Kinematics describes motion. Dynamics explains it. Three laws published by Isaac Newton in 1687 connect the forces on an object to the way it moves, and they remain the working foundation of mechanical, civil and aerospace engineering. Every crane rating, elevator cable, vehicle suspension and conveyor drive is designed with them.

This chapter states the three laws, introduces the forces you will meet most often (weight, normal force, tension, friction), and develops the single most important skill in mechanics: drawing a free-body diagram and turning it into equations. You will then apply the method to inclines, connected bodies, elevators and vehicles on curves.

Force and the first law

A force is a push or pull exerted on an object by something else. It is a vector, measured in newtons (N). One newton is the force that gives a 1 kg mass an acceleration of 1 m/s². For a sense of scale, a typical apple weighs about 1 N and an adult about 700 N.

Newton's first law: an object stays at rest, or keeps moving in a straight line at constant speed, unless a net force acts on it. Motion does not need a force to keep it going; change of motion does. A hockey puck slows down not because "its force runs out" but because friction acts on it.

The first law also defines what we mean by an inertial reference frame: one in which the law holds. The ground is inertial to a very good approximation for engineering purposes. An accelerating car is not, which is why a coffee cup on the dashboard appears to slide backward "by itself" when you accelerate.

A body is in equilibrium when the net force on it is zero. It may be at rest (statics) or moving at constant velocity, such as an aircraft in steady level flight, where thrust balances drag and lift balances weight.

The second law

When the net force is not zero, the object accelerates in the direction of the net force, by an amount inversely proportional to its mass:

This is a vector equation, so in practice it is two (or three) scalar equations, one per axis: \(\sum F_x = ma_x\) and \(\sum F_y = ma_y\). Mass \(m\), in kilograms, measures inertia: how hard it is to change an object's motion. Pushing a loaded cart is harder than an empty one because its mass is larger, not because it is heavier (the push is horizontal and weight is vertical).

Weight and mass

The weight of an object is the gravitational force the Earth exerts on it: \(W = mg\), directed downward, with \(g = 9.81\) m/s². A 1000 kg car weighs 9810 N. Mass is the same everywhere; weight depends on the local gravity. On the Moon (\(g = 1.62\) m/s²) the same car weighs 1620 N.

The third law

Newton's third law: when object A exerts a force on object B, B exerts a force on A of equal size and opposite direction. Forces always come in pairs.

The two forces in a pair act on different objects, so they never cancel each other in the equation for a single object. When a tow truck pulls a car, the cable pulls the car forward and the car pulls the cable backward with an equal force. The truck still accelerates because it also has a large forward friction force from the road on its driven wheels. Whether something accelerates depends only on the forces acting on it.

A common trap: the weight of a book on a table and the normal force from the table are equal and opposite, but they are not a third-law pair. They act on the same object (the book). The third-law partner of the book's weight is the book's gravitational pull on the Earth.

Common forces

• Weight \(mg\): always present near the Earth, straight down.
• Normal force \(N\): the push of a surface, perpendicular to it. It is whatever size it needs to be to stop the object passing through the surface, which is not always \(mg\).
• Tension \(T\): the pull of a rope, cable or chain, along it, away from the object. An ideal (light) rope has the same tension throughout, and an ideal pulley changes its direction without changing its size.
• Friction: the force a surface exerts parallel to itself, opposing sliding (or attempted sliding).
• Spring force: \(F = kx\), opposing the stretch or compression \(x\) (Hooke's law).

Friction

Friction is surprisingly well described by two simple rules. Static friction acts when surfaces are not sliding; it adjusts to whatever is needed to prevent motion, up to a maximum:

Once sliding starts, kinetic friction takes over, with a roughly constant value \(\mu_k N\), and \(\mu_k\) is usually smaller than \(\mu_s\). That is why it takes more force to start a heavy crate moving than to keep it moving, and why anti-lock brakes, which keep tires just short of skidding, stop a car faster than locked wheels. Typical values: rubber on dry concrete \(\mu_s \approx 1.0\), \(\mu_k \approx 0.8\); steel on steel (dry) \(\mu_s \approx 0.7\), \(\mu_k \approx 0.5\); greased steel \(\mu \approx 0.1\).

Free-body diagrams: the method

Almost every dynamics problem is solved with the same five steps:

• Choose the object (or system) and isolate it in a sketch.
• Draw every force acting on it, and only those: weight, then one force at every point where something touches it.
• Choose axes. Align one axis with the acceleration if you know its direction; on an incline, use axes along and perpendicular to the slope.
• Write \(\sum F_x = ma_x\) and \(\sum F_y = ma_y\).
• Solve, then check units, signs and limiting cases.

Elevator cables are sized for the peak tension during acceleration, plus large safety factors. The same reasoning applies to crane hoists and mine cages.

Inclined planes

With axes along the slope (\(x\), downhill positive) and perpendicular to it (\(y\)), the weight splits into \(mg\sin\theta\) along the slope and \(mg\cos\theta\) into it. The block does not accelerate perpendicular to the slope, so \(N = mg\cos\theta\).

That last result gives a quick way to measure \(\mu_s\): tilt a surface until the object just begins to slide and take the tangent of the angle. It is also the idea behind the angle of repose of gravel and soil stockpiles in geotechnical design.

Forces at an angle

Connected bodies

When objects are linked by ropes or in contact, draw a separate free-body diagram for each, then solve the equations together. The link gives a constraint: for an inextensible rope, the connected objects have the same magnitude of acceleration.

Circular motion

An object moving in a circle at constant speed is accelerating, because its direction keeps changing. The acceleration points toward the centre (it is centripetal) and has magnitude

By the second law, a net inward force \(mv^2/r\) is needed to keep it on the circle. That force is not a new kind of force: it is provided by friction, tension, a normal force, gravity, or a combination. Draw the real forces and set their inward sum equal to \(mv^2/r\).