• Unit 4: Dynamics

Kinematics is the study of motion. It describes the way objects move, their velocity, and their acceleration. Dynamics consider the forces that affect the motion of moving objects. Newton's Laws of Motion are the foundation of classical dynamics. These laws provide examples of the breadth and simplicity of principles under which nature functions.

Completing this unit should take you approximately 8 hours.

• 4.1: Newton's First Law of Motion

Newton's First Law of Motion is also called the Law of Inertia: an object at rest will remain at rest unless an outside force acts upon it. Also, an object in motion with constant velocity will remain in motion with constant velocity unless an outside force acts upon it. The reason for this law is to exemplify the functions of mass.

• 4.2: Newton's Second Law of Motion

As we saw in Newton's First Law of Motion, an object at rest stays at rest unless acted upon by an external force. Also, an object in motion at constant velocity remains in motion unless acted upon by an external force. Again, this is inertia. The only way to overcome inertia is to accelerate the object. Applying a net force to the system to induce acceleration.

Acceleration is proportional to the net external force on a system. That is, the higher the applied force, the bigger the acceleration. We also know that acceleration is inversely proportional to mass. That is, large objects accelerate at a slower rate than smaller objects. We know this from our everyday observations. It is easier to accelerate a light ball than a heavier bowling ball.

Newton's Second Law of Motion relates net external force to acceleration and mass of the system: $F_{\mathrm{net}} = ma$, where $F_{\mathrm{net}}$ is the net force, $m$ is mass, and $a$ is acceleration. Note that force is a vector quantity, so it has a magnitude and a direction.

The system is whatever we are interested in when calculating a physics problem. The external force is any force that acts upon the system, but is not part of the system. For example, picture pushing a rock up a hill. The system is the rock, and the external force is you pushing the rock.

The unit for force is the Newton, N. The definition of the Newton is $1\ \mathrm{N} = 1\ \mathrm{kg\: m/s^{2}}$.

• 4.3: Free-Body Diagrams

A free-body diagram shows all of the forces acting upon a system. It is a simplified way to visualize what is happening during a physics problem. Drawing the forces as vector arrows in the direction of the force from the center of the system can help us figure out how we need to add or subtract force vectors when determining the net force on an object.

• 4.4: Newton's Third Law of Motion

Newton's Third Law of Motion states that for every force exerted by an object to another object, there is an equal magnitude force exerted on the first object by the second object in the opposite direction. Some call this the Law of Action and Reaction.

In other words, for every action (exerted force), there must be an equal and opposite reaction. This law tells us that forces are always paired. Keep in mind that these forces act on two separate objects in pairs. The forces do not act on the same object being pushed.

• 4.5: Solving Problems Using Newton's Second Law: Weight

There are four common classical forces that will be discussed in the following sections: weight, normal force, tension, and friction. We will discuss each of these forces one at-a-time in each of the following sections.

Weight refers to the force of gravity on an object of a given mass. Because it comes from gravity, the weight force is generally directed toward the earth. The equation that relates the mass of an object to its weight is $F_{g}=mg$. This equation works only on or near the surface of the Earth.

Let's consider a coffee cup sitting on a table. The coffee cup is experiencing the force of its weight that draws it toward the center of the earth. This "pushes" down on the table. However, because of Newton's Third Law of Motion, there must be an equal magnitude force in the opposite direction also acting on this table to balance the forces.

• 4.6: Newton's Law of Gravity

Weight, in a more general sense, can be given using Newton's Universal Law of Gravitation. This law states that all objects in the universe attract each other in straight force lines between them.

• 4.7: Solving Problems Using Newton's Second Law: Normal Force

Otherwise known as the "force of contact", we have the normal force. Here, "normal" essentially means perpendicular. That is, we give it the name normal force to make it obvious that it points perpendicularly to a surface.

For example, normal force balances the weight from a cup of coffee on a table and keeps it from going through the table. This is why we often call normal force the force of contact. In other words, we can say that normal force is the force that prevents two objects from being in the same place. The normal force is often abbreviated as N. Do not confuse the symbol for normal force as the unit of force, Newtons.

• 4.8: Solving Problems Using Newton's Second Law: Tension

Tension is the force along the length of an object. We normally think of tension as a force of the object's strength, such as for a rope. Objects, such as ropes, can only exert forces in the same direction as their length. If a rope is attached to a hanging object, the object's weight exerts a force toward the earth while the rope acts as a tension force in the opposite direction of the weight. Much like the normal force from the table mentioned earlier, this keeps the object from falling down.

• 4.9: Solving Problems Using Newton's Second Law: Friction

Friction is the force between two surfaces in contact that opposes parallel motion between them. Kinetic friction is the friction between surfaces that are moving or sliding relative to each other. Static friction is the friction that occurs that prevents two surfaces from moving or sliding with respect to each other.

Static friction varies based on how much counter force is needed to prevent two objects from sliding. When a certain amount of force is applied to a stationary object in contact with a surface, static friction serves to counter that applied force in order to keep the object and surface in contact from sliding. The more force that is applied, the more static friction is summoned to counter it.

However, static friction is only so strong. So once the maximum static frictional force is summoned, the object will start to slide and the static friction force will give way to kinetic friction force. Generally, the maximum ability of static friction is higher than the maximum ability of kinetic friction. That is, once the maximum static friction force is met, the object undergoing applied force will jolt forward because the kinetic friction that took over is weaker than the maximum static friction force that held it previously.

Kinetic and static frictional forces are given by the equations:

$F_{fr,k}=\mu _{k}F_{N}$
$F_{fr,s}\leq \mu _{s}F_{N}$

Note that the static friction force equation is an inequality. That is because static friction only aims to counter potential movement or sliding between two surfaces, which vary based on the magnitude of the applied force. Both equations have a $\mu$ symbol which is called the coefficient of friction and depends solely on the types of materials that make up the two surfaces. For example, between steel and ice, the coefficient of friction would be very small (perhaps 0.1). However, between rubber and concrete, the coefficient of friction would be rather large (perhaps 0.8). The coefficient of friction is rarely larger than one.

We experience friction often in our everyday lives. For example, if you slide a box across a room, the box's motion will eventually stop due to the friction that occurs between the surface of the box and the surface of the floor. A box will slide relatively well across a smooth tile floor because the smooth tile floor provides a lower frictional force. It will slide less well across a floor with a rough carpet because the carpet provides a higher frictional force.

When we walk on a sidewalk our shoes do not generally slip because the static friction between our shoes and the sidewalk opposes the forward force of our shoes. However, we know that icy surfaces are "slippery" when the ice exerts less friction on our shoes than concrete.