Newton’s Laws and FBDs

Many of you have heard of this scientist called Isaac Newton. Perhaps you have even heard that he wrote a few laws. Well, that’s what this section is about, but first we need to understand what force is.

Force

Whenever we interact with the world, we have to apply a force on something in order to move or stop it. Our legs apply a force on the ground and car axles apply a force on the wheels to create motion.

Newton’s three laws describe motion in inertial frames within classical mechanics. They are excellent models for ordinary speeds and scales.

Newton’s Laws of Motion

Newton’s First Law: In an inertial frame, an object stays at rest or moves with constant velocity if the net external force is zero. A change in velocity requires a nonzero net external force.

Newton’s Second Law: In an inertial frame, the net external force is the rate of change of momentum. For constant mass, it equals mass times acceleration. This applies even when acceleration varies with time.

ΣF⃗ = ma⃗ (Equation 1.3.1). Force is measured in newtons: 1 N = 1 kg·m/s². If only one force acts, it is also the net force.

Newton’s Third Law: When A exerts a force on B, B exerts an equal and opposite force on A. The two forces act on different objects; they do not cancel in one object’s free-body diagram. See Gallery 1.3.2.

Free Body Diagrams

Free body diagrams (FBDs) are used for analyzing the forces acting on an object. They are important because they help us calculate the net force acting on an object. Figure 1.3.1 shows a very simple free body diagram. Let’s assume that this is a ball in mid-air on Earth and let’s also assume that there is no air resistance. Therefore, there is only one force acting on it and it is directed downward as displayed. Recall that Newton’s Second Law states that $\overrightarrow{F}=m\overrightarrow{a}$, where any object that has a mass and acceleration will have an associated force with it.

The force on this ball, as shown in the picture, is $\overrightarrow{F}=m\overrightarrow{g}$. The $\overrightarrow{g}$ in the equation represents gravity↗. It has a constant magnitude and is always directed downwards in all FBDs. It is represented numerically by having a value of $9.81\text{m}/{\text{s}}^2$ downward. In our example, gravity ($\overrightarrow{g}$) replaces acceleration ($\overrightarrow{a}$) in Equation 1.3.1. Gravity in this case, is the source of acceleration that is acting on the ball and is what causes it to move in the downward direction. It is referred to as the force of gravity and can be written as ${\overrightarrow{F}}_g$_.

${\overrightarrow{F}}_g=m\overrightarrow{g}$ (Equation 1.3.2)​

Does changing the ball’s initial velocity change its weight? In this near-Earth model, no: F⃗_g = mg⃗ is independent of velocity. Other forces, such as air drag, can depend on velocity.

Figure 1.3.2 shows a constant horizontal force applied to a box on a frictionless plane. The box accelerates according to Newton’s second law. The third-law partner is the force the box exerts back on the agent applying the push.

That is not the only force acting on the block, though, so our Free-Body Diagram is incomplete. Have you ever wondered how you are able stand on the ground without falling through? Or how your table doesn’t break when you place a lot of textbooks on top of it? This is because the ground is pushing back on you with a force just enough to prevent you from having any vertical acceleration. This is called the Normal Forceⓘ, which can be written as either ${\overrightarrow{F}}_N$ or $\overrightarrow{N}$. It is defined as the force exerted on an object which is perpendicular to the plane of contact. It is another ever-present force to include in your FBDs. In the box in Figure 1.3.2, the normal force would be acting directly upward on the box to counteract gravity .

Figure 1.3.3 shows the full free body diagram for our box system. Notice how the forces are clearly labeled with arrows to indicate the direction that those forces are acting. The FBD shows that there is an applied force ${\overrightarrow{F}}_{app}$ acting on the box, a force due to gravity, $m\overrightarrow{g}$ and a normal force ${\overrightarrow{F}}_N$. This is a complete free body diagram.

Inclined Plane with Friction

Next we are going to discuss another classic system. Instead of a flat plane, we place a box on an inclined plane, and we still have a force applied to it, this time pulling it to the left. We will also introduce friction here. With friction, as the box slides down the inclined plane, it should experience some resistance in the opposite direction of motion. This resistance that the box will feel as it slides down the plane is known as the Friction Force or ${\overrightarrow{F}}_{fr}$.

Examples of this can be when you rub your hands together and you feel the resistance in between them or when you slide in your socks on the floor.

Key Point: Kinetic friction opposes sliding between the contacting surfaces. Static friction opposes their tendency to slide; it need not oppose the object’s motion relative to the ground.

Figure 1.3.4 is the free body diagram for our box on an inclined plane. Notice again how the FBD only shows the forces acting on the object that affects its overall motion.

In our FBD, there are four forces affecting the object. The applied force $\overrightarrow{F}$, the gravitational force $m\overrightarrow{g}$, the normal force $\overrightarrow{N}$, and the frictional force ${\overrightarrow{F}}_{fr}$.

An important thing to notice is that up to this point, we’ve only been discussing forces and motion of objects on flat surfaces. With this inclined plane example, notice how the gravitational force and normal force are not in the same direction. Since force is a vector, it has components, which we will be something that we cover later. Just keep this concept of components in mind .

Key Point: The Normal Force always acts perpendicular to the plane on which an object is. It is the force exerted by the dirt underneath your feet that keeps you from sinking into the ground. It is the combination of forces exerted by your hand that allows you to hold this device steady. It takes on many forms.