We have already discussed the acceleration of an object caused by gravitational attraction. From Newton’s Second Law of Motion, we know that the force is the product of the object’s mass and its acceleration, so that the force of gravity can be written as \(F_{\text{g}} = mg\text{,}\) where \(g\) is the acceleration from gravity for the problem (earth gravity is \(g = 9.8 \text{m}/\text{s}^2\)).

The free body diagram for this on its own is just a single arrow pointing doward, indicating the force of gravity on the object. As it stands, this object is experiencing a net force, and so it is accelerating downward. If there’s nothing to apply an upward force, the object will continue to accelerate, causing it to fall faster and faster.

We are going to add some air resistance to the gravity example. Now as the object is falling, it’s also experiencing air resistance. This can be thought of as the impact of the object bumping into molecules of as it falls. As it falls faster, it bumps into more air, and so it experiences more resistance.

The free body diagram now has two forces. Gravity is still pulling it downward, but now there’s also an upward force that we can call \(F_{\text{air}}\text{.}\) The size of this force will depend on a number of factors, such as the weight and shape of the object. An object that is more streamline in the direction of motion will hit fewer air molecules than something that is bulky in that direction. But regardless of the shape, there is some amount of upward force from the air.

There is a specific when the force of gravity perfectly matches the force of air resistance. This is known as terminal velocity. When an object is falling at terminal velocity, there is zero net force on the object because the two forces are perfectly cancelling each other out. This means that there is no acceleration, which means that the object’s motion is unchanging (Newton’s First Law), and the object will remain at this velocity until something changes.

One of the most challenging types of forces to understand is the normal force. This is the force that an object imparts on another object when they interact with each other. This force is always viewed as being perpendicular to the surface.("Normal" is a mathematical term for perpendicularity.) Such forces always exist whenever two objects are physically interacting with each other.

Consider a ball sitting on a table. If we tried to apply Newton’s First Law, we would see an object at rest that is staying at rest, and so we know that there is no net force acting on it. However, since this ball is subject to gravity, and gravity is a downward force, in our force diagram we must have a downward arrow to represent that. If this were the only force acting on the ball, then the ball would start to accelerate towards the ground. But the ball is at rest, and that’s because the table is applying an upward force to the ball. And since the ball is at rest, the upward force of the table must be exactly equal to the downard force of gravity.

We can also see how this normal force works to create motion. Instead of thinking of the ball on a table, let’s think about it on an incline. Remember that the normal force is always perpendicular to the surface, so that the incline does not apply a vertical force to the ball. When we draw the force diagram, we will have gravity pulling the ball downward, but then the normal force would push up at an angle. Notice that the angle of the normal force indicates which direction (left or right) the ball is going to roll. In fact, we can know that the net force the ball is going to experience must be parallel to the incline because that’s the ultimate direction the ball will move. From this, we can do a vector analysis to determine exactly how large the normal force must be.

Friction is the force that resists the relative movement of objects sliding against each other. Friction can occur between many types of objects: solid objects against other solid objects, solid objects against a fluid (such as air or water resistance), and a fluid against itself are just some of the ways friction can be manifested.

We are going to focus our attention on the friction between solid objects, which is known as surface friction. But even this isn’t quite an entirely simple concept, because there are two types of friction. There’s static friction, which is the friction between two objects when there is no movement between them, and then there’s kinetic friction, which is the friction between two objects when they are moving relative to each other.

The formula for the force of friction between two objects at rest relative to each other is \(F \leq \mu_s N\text{,}\) where \(N\) is the magnitude of the normal force and \(\mu_s\) is the coefficient of static friction, whose value depends on the two surfaces that are in contact with each other. The direction of the friction force is always opposite of the direction that resists movement. Smaller values of \(\mu_s\) mean that there is less friction, with 0 being no friction at all. The reason that the force is given as an inequality is that the quantity on the right represents the maximum amount of friction that can exist between the two surfaces.

The coefficient of kinetic friction (denoted \(\mu_k\)) plays the same role as the coefficient of static friction, except it applies when the two objects are sliding past each other. The coefficient of kinetic friction is always smaller than the coefficient of static friction for the same pair of objects. You have experienced this if you have ever tried to move something heavy. It takes a certain amount of effort to get it moving, and once it’s moving it takes less effort to keep it moving. In this case, the formula is \(F = \mu_k N\text{,}\) and this is an equality because in this situation the friction is always at its maximal value. If the movement of the object slows down to the point that it is no longer moving, then we would be back in the case of static friction.