Forces, work and energy

Forces belong to interactions

Pushing a box, stretching a spring and attracting an object through gravity are interactions. A force is a vector quantity representing one of those interactions on a body. It has magnitude, direction and sense, and is measured in newtons, N. To identify it, ask: which body exerts this force, and on which body does it act?

Some forces require contact, such as a table's force on a book. Others, such as gravity, act without direct contact. A body can experience several forces at once. Net force is the vector sum of all external forces on the chosen body. Opposite forces of 12 N and 7 N have a resultant of 5 N in the larger force's direction, not 19 N.

Draw a diagram of only the body under study, representing each force with a labelled arrow. Do not add a “force of motion”: moving is not an interaction. Nor should you draw on the body the force it exerts on another; that force belongs in the other body's diagram.

Net force changes velocity

In an approximately inertial frame, a body with zero net force keeps constant velocity: it can remain at rest or move in a straight line at constant speed. This is Newton's first law. The habit of thinking “you must push to keep moving” comes from situations where friction is always present.

For a constant-mass body, the second law is ΣF = ma. Along a line, signs indicate directions. Mass m is measured in kg and acceleration a in m/s²; therefore 1 N = 1 kg·m/s². The same net force produces smaller acceleration for a larger mass.

The law relates force to acceleration, not directly to velocity. A body can travel rightward with a leftward net force: at that instant its speed decreases. If the net force is perpendicular to velocity, it changes motion direction. Before looking for numbers, predict what change in motion the forces allow.

Mass, weight, support and pairs

Mass is a body property associated with inertia. Weight is the gravitational force exerted on it by an astronomical body. P = mg near Earth's surface, with g ≈ 9,8 m/s² in this lesson. Weight is measured in N; mass in kg. Changing planets alters g and weight without altering the same body's mass.

The normal force is a contact force perpendicular to the surface. For a book resting on a horizontal table with no other vertical forces or vertical acceleration, N − mg = 0, hence N = mg. This equality follows from these conditions. On an inclined support, in an accelerating elevator or when someone pulls the object vertically, the forces must be analyzed again.

The third law says that if A exerts a force on B, B simultaneously exerts on A a force of equal magnitude and direction but opposite sense. The forces act on different bodies. A book's weight and normal force are not an action–reaction pair because both act on the book. The partner of the book's weight is the book's gravitational attraction on Earth.

Work is a way to transfer energy

A force can transfer energy when its point of application moves. For constant force and straight-line displacement, work is W = Fd cos θ. F is force magnitude in N; d displacement magnitude in m; θ the angle between the two vectors. The unit is joule, J = N·m. In this introduction we mainly use forces parallel or perpendicular to displacement.

If force and displacement point the same way, W = +Fd; if opposite, W = −Fd. If perpendicular, W = 0. Holding a stationary box can tire muscles, but the force on the box does no mechanical work on it while the box does not move. The body can use energy in other processes.

Work belongs to a force during a process; it is not something stored in the object. Calculate each force's work separately and add them to obtain net work. For a box sliding horizontally, weight and normal force can do zero work while pulling does positive work and friction negative work.

Kinetic energy and potential energy

Translational kinetic energy is K = mv²/2, with m in kg and speed v in m/s. It is measured in joules and does not depend on motion direction. Doubling mass doubles K, but doubling speed quadruples K. This dependence explains why a small speed increase can require much more energy to stop a vehicle.

The work–energy theorem states W_net = K_f − K_i. The sum of work determines the change in kinetic energy. An individual force can do positive work without K increasing: another force can remove the same amount. Constant velocity means zero change in K, not necessarily no transfers.

Near Earth's surface, gravitational potential energy of the body–Earth system can be written U = mgh, choosing h = 0 at a reference level. The difference ΔU = mgΔh is what affects transfers. Raising the zero level changes U values without changing differences or observable results.

Conserving energy requires choosing the system

When only gravity changes a falling body's kinetic energy, the increase in K equals the decrease in U. Mechanical energy E_m = K + U is conserved in this model. If air resistance exists, some mechanical energy becomes internal energy of the body and air; total energy does not disappear.

Choosing the system helps organize the explanation. For the body alone, gravity is an external interaction that can do work. For the body–Earth system, gravitational interaction is internal and we track U. Both descriptions should lead to the same motion if used consistently.

Average power is energy transferred per duration: P_m = E/Δt, in watts, W = J/s. Two motors can transfer the same energy in different times and have different powers. The letter P can also mean weight in some contexts; always read the definition and unit. A more powerful machine does not create energy: it can transfer it faster.

Simple machines: trading force for displacement

A simple machine can help lift a load by applying a smaller force over a greater displacement. It does not create energy. In an ideal lossless model with parts starting and finishing at the same kinetic energy, supplied work equals useful work: input F × input d = output F × output d when forces are constant and parallel to the displacements considered.

Mechanical advantage is the ratio of magnitudes output F/input F. An advantage greater than 1 does not mean efficiency above 100%: input displacement compensates for the force gain. On a frictionless incline, raising weight mg through height h requires energy mgh. If the path along the ramp is L and speed is constant, the required parallel force is F = mgh/L. Increasing L reduces F but increases the distance over which force is exerted.

For example, raising a load weighing 120 N through 0,50 m requires 60 J. On an ideal 2,0 m ramp, only 30 N parallel to the path is needed, because 30 × 2,0 = 60 J. Force is four times smaller and path four times longer. With friction, supplied work is greater than the potential-energy increase; the difference appears in other transfers, such as heating.

For a lever in equilibrium, rotational effect also matters. The magnitude of a force's moment about the support is force times perpendicular lever arm. If opposite rotational effects balance, input F × input arm = load F × load arm. A 180 N load with arm 0,20 m balances a 60 N force with arm 0,60 m: both moments have magnitude 36 N·m. The arm is the perpendicular distance to the line of action, not automatically the bar's total length.

Pulleys, wheels and other machines can also change directions and relationships between force and displacement. Before using a formula, identify the machine, forces and assumptions. Efficiency is useful work/supplied work; losses make it lower than the ideal model. Comparing work preserves the physical meaning of “gaining force” without promising free energy.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Mass and weight are names for the same thing.

Mass is measured in kg; weight is gravitational force, measured in N.

Normal force is always mg.

Find normal force from the forces and acceleration perpendicular to the support.

The work of any force is the change in kinetic energy.

The theorem uses net work, meaning the sum of work.

Energy is lost in friction.

Mechanical energy decreases and internal energy increases; total energy remains conserved.