Like a shepherd who brings back stray sheep, we need to rescue the word force and give it precise meaning. In physics force means something very specific. Not “the force” from Star Wars, not the “force of public opinion”, and not the force in the battle of good versus evil.
Force in physics has a precise meaning as an amount of push or pull exerted on an object. Forces are vector quantities measured in Newtons [N]. In this section we will explore all the different kinds of forces.
We next list all the forces which you are supposed to know about for a standard physics class and define the relevant parameters for each kind of force. You need to practice exercises using each of these forces, until you start to feel how they act.
The force of gravity exists between any two massive objects. The magnitude of the gravitational force between two objects of mass $M$[kg] and $m$[kg] separated by a distance $r$[m] is given by the formula \[ F_g=\frac{GMm}{r^2}, \] where $G=6.67 \times 10^{-11}$[$\frac{\text{Nm}^2}{\text{kg}^2}$] is the gravitational constant. This is the famous one-over-arr-squared law that describes the gravitational pull between two objects. This was Newton's big discovery.
On the surface of the earth, which has mass $M=5.972\times 10^{24}$[kg] and radius $r=6.367\times10^6$[m], the force of gravity on an object of mass $m$ is given by \[ F_g=\frac{GMm}{r^2} = \underbrace{\frac{GM}{r^2}}_{g}m = 9.81 m = W. \] We call this force the weight of the object and to be precise we should write $\vec{W}=-mg\hat{\jmath}$ to indicate that the force acts downwards—in the negative $y$ direction. Verify using your calculator that $\frac{GM}{r^2}=9.81\equiv g$.
A spring is a piece of metal twisted into a coil that has a certain natural length. The spring will resist any attempts to stretch it or compress it. The force exerted by a spring is given by \[ \vec{F}_s=-k\vec{x}, \] where $x$ is the amount by which the spring is displaced from its natural length and the constant $k$[N/m] is a measure of the strength of the spring. Note the negative sign: if you try to stretch the spring (positive $x$) then the force of a spring will pull against you (in the negative $x$ direction), if you try to compress the spring (negative $x$) it will push back against you (in the positive $x$ direction).
The normal force is the force between two surfaces in contact. The word normal means “perpendicular to the surface of” in this context. The reason why my coffee mug does not fall to the floor right now is that the table exerts a normal force $\vec{N}$ on it keeping in place.
In addition to the normal force between surfaces, there is also the force of friction $\vec{F}_f$ which acts to prevent or slow down any sliding motion between the surfaces. There are two kinds of force of friction and both kinds of are proportional to the amount of normal force between the surfaces: \[ \max \{ \vec{F}_{fs} \}=\mu_s\|\vec{N}\| \ \ \text{(static)}, \qquad \vec{F}_{fk}=\mu_k\|\vec{N}\| \ \ \text{(kinetic)}, \] where $\mu_s$ and $\mu_k$ are the static and dynamic friction coefficients. Note that it makes intuitive sense that the force of friction should be proportional to the magnitude of the normal force $\|\vec{N}\|$: the harder the surfaces push against each other the more difficult it should be to make them slide. The above equations make this intuition precise.
The static force of friction acts on objects that are not moving. It describes the maximum amount of friction that can exist between two objects. If a horizontal force greater than $F_{fs} = \mu_s N$ is applied to the object, then it will start to slip. The kinetic force of friction acts when two objects are sliding relative to each other. It always acts in the direction opposite to the motion.
A force can also be exerted on an object remotely by attaching a rope to the object. The force exerted on the object will be equal to the tension in the rope $\vec{T}$. Note that tension always pulls away from an object: you can't push a dog on a leash.
Viewing the interactions between objects in terms of the forces that act between them is a very powerful way of thinking. In the next section, we will learn how to draw force diagrams which take into account all the forces that act on the object.