Force and Motion

Kinematics: describing motion before explaining it

Mechanics splits into two questions. Kinematics asks how a body moves; dynamics asks why. Doing them in that order matters, because you cannot check a force analysis until you can say precisely what motion it is supposed to produce.

Position is where a body is, measured from an origin you choose. Displacement is the change in position, the vector from start point to end point. Distance traveled is the length of the path actually followed: a scalar, never negative, and never smaller than the magnitude of the displacement. A car that leaves a parking space, drives eight kilometers of city streets, and returns to the same space has traveled eight kilometers and been displaced by zero.

Velocity is the rate of change of position and inherits the vector character of displacement. Average velocity is total displacement divided by elapsed time; instantaneous velocity is the limit of that ratio as the time interval shrinks. Speed is the magnitude of instantaneous velocity, so it has size but no direction. The distinction survives at the average level too: average speed is total distance over elapsed time, while average velocity is total displacement over elapsed time, and the two are not generally the same number. The car above had a healthy average speed and an average velocity of exactly zero.

Acceleration is the rate of change of velocity. Because velocity is a vector, a body accelerates whenever its speed changes, its direction changes, or both. A car holding a steady 50 km/h around a curve is accelerating, and that is not a verbal trick: a real sideways force from the tires is producing it.

The constant-acceleration equations and the assumption inside them

When acceleration is constant, calculus collapses into algebra and you get the familiar set: v = v0 + a t; x = x0 + v0 t + (1/2) a t squared; v squared = v0 squared + 2 a (x - x0); and x = x0 + v-bar t, where the average velocity v-bar is (v0 + v)/2. Any three of the five quantities — initial velocity, final velocity, acceleration, elapsed time, and displacement — fix the other two.

The assumption is doing real work, and it is easy to forget that it is there. These equations hold only while acceleration is constant in both magnitude and direction. They do not describe a rocket whose mass falls as it burns propellant, a skydiver once air resistance has grown with speed, or any stretch of motion that spans a change in the forces acting. Constant acceleration is a sound model when the net force and the mass are both constant — exactly the situation of a block sliding under a steady push against steady friction, and exactly not the situation of most real machines across a full cycle. When the forces change, either break the motion into intervals over which they are effectively constant, or integrate.

One habit protects you here: choose a positive direction before writing any equation and hold it for the whole problem, so that a negative answer means something. Treat one-dimensional motion as vector motion carrying a sign, not as scalar motion.

Newton's three laws, stated carefully

First law. A body at rest stays at rest, and a body in motion continues at constant velocity, unless a net external force acts on it. Read as a claim about objects this looks like nothing more than the second law with zero on the right-hand side, which is why the more useful reading is the other one: the first law tells you which reference frames the rest of Newtonian mechanics is entitled to use. A frame in which an isolated body really does keep constant velocity is an inertial frame. Any frame moving at constant velocity relative to an inertial frame is also inertial; an accelerating or rotating frame is not, and in such a frame the second law fails unless you add correction terms. Earth's surface is close enough to inertial for essentially every problem in a first course, and not close enough for problems in which Earth's own rotation matters.

Second law. In an inertial frame, the acceleration of a body is proportional to the net external force on it and inversely proportional to its mass: acceleration equals net force divided by mass, usually written as the sum of forces equals mass times acceleration. Three details carry weight. The sum runs over external forces only, because internal forces between parts of the chosen body cancel in pairs and cannot accelerate it. The statement is a vector equation, which in practice means one scalar equation per axis. And the mass-times-acceleration form is the constant-mass special case: Newton's general statement is that net force equals the rate of change of momentum. For a rocket, a hopper being filled on a moving belt, or any system whose mass changes while it moves, the momentum form is the one that still holds.

Third law. If body A exerts a force on body B, then B exerts on A a force of equal magnitude and opposite direction. The half of this that students lose is the important half: the two forces act on different bodies. They therefore never cancel, because cancellation happens only among forces acting on the same body, and the two halves of a third-law pair by construction never appear in the same net-force sum. When you push on a wall, your push acts on the wall and the wall's push acts on you; that neither of you accelerates much is down to the other forces acting on each of you, not to the pair canceling. When a swimmer pushes off the end of a pool, the wall's reaction is the only forward force she gets, and she accelerates precisely because that force acts on her while hers acts on the wall.

Mass, weight, and the newton

Mass measures inertia: how much force a given acceleration costs. Its SI unit is the kilogram, and it does not change when you move the body. Weight, in scientific and engineering usage, is a force: the force that would give the body an acceleration equal to the local acceleration of free fall. Its unit is the newton, and it does change with location. NIST's guide to the SI is direct about the resulting trap: in everyday and commercial speech, weight means mass and is quoted in kilograms, so technical writing should say mass when it means mass, and should make the intended sense explicit on any occasion when it says weight.

The newton is the SI coherent derived unit of force, equal to one meter kilogram per second squared — one newton gives one kilogram an acceleration of one meter per second squared. Where U.S. customary units are conventional, force is quoted in pounds-force, and NIST's conversion factor is one pound-force equals 4.448 222 newtons; the obsolete kilogram-force equals 9.806 65 newtons exactly.

Then there is g. It is an acceleration, not a force, and not a property of the object: near Earth's surface a body in free fall accelerates at roughly 9.81 meters per second squared, and the weight of a body of mass m is w = m g. Two cautions follow. Local g varies across Earth's surface, so weight is genuinely location-dependent — on the Moon, where g is about 1.62 meters per second squared, a one-kilogram mass weighs about 1.6 newtons instead of about 9.8. And the standard acceleration of gravity, 9.806 65 meters per second squared, is a defined conventional value adopted by the 3rd General Conference on Weights and Measures in 1901 and listed by NIST as exact. Use it for defined conversions; do not treat it as a measurement of the field in the room you are standing in.

Force types, net force, and equilibrium

A force is a push or a pull that one body exerts on another, which means every force in a Newtonian problem has an identifiable source body. Five kinds cover most of an introductory course.

Gravitational force is Earth's attraction on the body, w = m g near the surface, directed toward Earth's center. Normal force is the contact force a surface exerts perpendicular to itself. It is a response force with no fixed value of its own: it takes whatever magnitude the rest of the problem demands, up to the load at which the surface fails. On a level floor with no vertical acceleration it happens to equal the weight; on an incline, or inside an accelerating lift, it does not. Tension is the force transmitted along a rope, cable, or chain, and it pulls along the line of the rope — it never pushes. Applied force is the catch-all for a push or pull whose source falls outside the other categories.

Friction is the contact force parallel to a surface, opposing sliding or the tendency to slide. The standard coefficient model is empirical rather than fundamental, as its own sources state. Static friction obeys an inequality — it is at most the static coefficient times the normal force — because static friction takes exactly the value needed to prevent sliding, right up to a maximum. Kinetic friction is modeled as an equality: the kinetic coefficient times the normal force, once sliding has begun. For a given pair of materials the kinetic coefficient is generally smaller than the static one, which is why a stuck crate lurches when it finally breaks free. The model also treats friction as independent of contact area and of sliding speed — approximations that fail at high speeds, at high contact pressures, and with lubrication, wear, or surface contamination.

Net force is the vector sum of every external force on the chosen body, and it is the only thing the second law responds to. Equilibrium is the special case in which that sum is zero, which by the first law means the body is at rest or moving at constant velocity — not necessarily at rest. Organizing that sum on paper, by isolating one body and accounting for every external force on it, is the free-body diagram, the next topic in this unit and where that technique is taught. The zero-sum case is statics; the non-zero case is dynamics; accumulating force over distance and over time gives work and energy, and momentum. All four build on the vocabulary set here.

Idealizations, and where each one breaks

Everything above is a model, and the models carry named simplifications. Say them out loud when you use them, because a model is not wrong for being idealized — it is wrong when it is used outside the range where its simplifications hold, and naming the assumption is what lets you check that.

Point mass, or particle: the body is treated as all of its mass concentrated at a point, so forces have no moment arms and the body cannot rotate. Fine for a crate sliding as a unit; it fails the moment rotation, tipping, or the location where a force is applied begins to matter.

Rigid body: distances between points within the body never change. A working approximation for a steel bracket under modest load, a poor one for a rubber mount, a long slender member, or any component near yield — deformation is the whole subject of mechanics of materials.

Massless rope and frictionless pulley: tension is then the same everywhere along the rope and unchanged around the pulley. Real cables have mass that matters over long spans, and real pulleys have bearing friction and rotational inertia that matter whenever accelerations are large.

Constant g: sound over the height of a building, wrong across the altitude range of a launch vehicle. Frictionless surface: a deliberate fiction used to isolate one effect, and nothing real is frictionless. Constant mass: assumed by the mass-times-acceleration form of the second law and violated by anything that burns, sheds, or accumulates material.

A closing note on scope. These lessons are educational material, not engineering design guidance. Real design of a structure, machine, pressure vessel, or electrical installation requires a licensed engineer working to the governing code, with verified material properties and margins chosen for the consequences of failure. Nothing here — including the assumed friction coefficients in the worked example — should be carried into a real design.