South African NSC Physical Sciences

Kinds of force

A force is a push or a pull, measured in newtons (N). It is a vector, so it has a magnitude and a direction, and two forces on the same object can only be combined by taking direction into account.

The forces met most often in mechanics are these.

Weight is calculated from the mass of the object and the gravitational acceleration at its position:

where $m$ is mass in kg and $g = 9.8\ \mathrm{m\cdot s^{-2}}$ near the surface of the Earth. Mass is the amount of matter in an object and does not change with position, whereas weight changes with the gravitational field the object is in.

Force diagrams and free-body diagrams

A force diagram shows the object and every force acting on it, drawn as arrows from the object, with the object's surroundings (a slope, a pulley, a second object) also drawn. A free-body diagram is a simpler diagram in which the body is isolated from its surroundings and drawn as a dot or a small box. Only the forces acting on that body are drawn, as arrows from the dot, and their lengths show their relative sizes. Each arrow is labelled with its symbol.

Some habits make a free-body diagram correct:

• Draw only forces that act on the body, never forces the body exerts on something else.
• Draw weight straight down, even on a slope.
• Draw the normal force perpendicular to the surface the body touches.
• Draw friction parallel to the surface and opposing the motion, or the way the body would move if friction were absent.
• Do not draw the resultant force as well as the forces it replaces, and do not draw a force for the acceleration or for the motion itself.

Resolving forces into components

Many problems have a force at an angle to the direction of motion. The force is split into two perpendicular components along chosen axes, and each axis is treated separately. A force $F$ at an angle $\theta$ to the $x$-axis has components

On an inclined plane the most useful axes are parallel and perpendicular to the plane, not horizontal and vertical. The weight of an object on a plane inclined at angle $\theta$ to the horizontal has the components

The parallel component pulls the object down the slope and the perpendicular component presses it into the surface. For a 5 kg block on a slope of $30^\circ$, $w = 5 \times 9.8 = 49\ \mathrm{N}$, so

If nothing else pushes the block into or away from the slope, the normal force equals $w_\perp$, so $N = 42.4\ \mathrm{N}$. The normal force is not always equal to the weight: it is smaller than $mg$ on a slope, and changes again if an applied force has a vertical component.

The net force

The net (resultant) force is the vector sum of all the forces on the object. In one dimension the forces along a line are added with a sign convention, for example right positive, so a 12 N force to the right and a 5 N force to the left give $12 - 5 = 7\ \mathrm{N}$ to the right. In two dimensions each axis is added separately, and the net force is found from its two components with Pythagoras' theorem and $\tan\theta = F_y / F_x$.

If the net force is zero the object is in equilibrium: it is at rest or moving at constant velocity. If it is not zero, the object accelerates in the direction of the net force.

Friction

Friction acts where two surfaces touch, and it is always parallel to the surfaces.

Static friction, $f_s$, is the force that opposes the tendency of a stationary object to move relative to a surface. It adjusts to match whatever force is trying to move the object, up to a limit. Kinetic friction, $f_k$, is the force that opposes the motion of an object that is moving relative to a surface.

The maximum static friction, $f_s^{max}$, is reached just before the object begins to move. Kinetic friction is usually a little smaller than this maximum.

Experiments show that friction:

• is proportional to the normal force, so pressing the surfaces together harder increases it;
• does not depend on the area of contact;
• does not depend on the speed of the motion (for kinetic friction).

Each relationship uses a coefficient that depends on the two surfaces, has no unit and is usually less than 1:

$\mu_s$ is the coefficient of static friction and $\mu_k$ the coefficient of kinetic friction.

Friction below and above the maximum

If a force $F$ parallel to the surface is applied to a stationary object and the object stays still, the static friction has exactly the same magnitude as $F$ and the opposite direction. Static friction is therefore not always $\mu_s N$: that value is only its ceiling. When the applied force exceeds $f_s^{max}$, there is a net force and the object accelerates, and from then on the friction on it is kinetic.

Worked example

A 20 kg crate rests on a horizontal floor with $\mu_s = 0.40$ and $\mu_k = 0.30$. The floor is level and no vertical force other than the weight acts, so

A horizontal push of 60 N is less than 78.4 N, so the crate does not move and the static friction is 60 N. A push of 100 N is more than 78.4 N, so the crate moves, and

Friction on a slope

On a rough slope the normal force is $mg\cos\theta$, so

Friction acts up the slope when the object slides or tends to slide down it. An object just begins to slide when $mg\sin\theta = \mu_s\, mg\cos\theta$, which gives $\tan\theta = \mu_s$, and mass cancels out, so the critical angle does not depend on how heavy the object is.