### 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.**

Upon successful completion of this unit, you will be able to:

- compare and contrast mass and inertia;
- determine the net force on an object;
- draw and interpret free-body diagrams representing the forces on an object;
- identify the correct use of normal and tension forces in terms of Newton's Third Law of Motion;
- use Newton's Second Law of Motion to analyze dynamic problems;
- apply Newton's Law of Gravity; and
- give examples of the effects of friction on the motion of an object.

### 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.

Read this text, which explains how we define mass as the amount of matter in an object. We measure mass in units, such as kilograms. Mass does not depend on the strength of the gravitational field and therefore does not depend on the location where it is being measured.

Inertia is the property of matter which mass quantifies. It describes the fact that an object at rest (not moving) will stay at rest unless an outside force acts upon it. Likewise, an object in motion will stay in motion with constant velocity unless an outside force acts upon it. These outside forces accelerate the object.This lecture accompanies what you just read.

This video provides a brief explanation on the thought process behind Newton's First Law and some historical context to the Law of Inertia.

This video provides a more detailed explanation behind the concepts of Newton's First Law.

### 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: , where is the net force, is mass, and 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 .This video reviews how Newton's Second Law was derived using concepts that are familiar from previous units.

As you read, pay attention to examples 4.1 and 4.2, which use Newton's Second Law of Motion to calculate acceleration and force in objects in motion.

This lecture accompanies what you just read.

This video discusses Newton's Second Law and solves a few sample problems. It delves into a more detailed analysis of solving for Force, Mass, and Acceleration of dynamic situations.

This video presents additional sample problems involving Newton's Second Law.

### 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.

Read this text to see examples of how to draw a free-body diagram like we saw earlier in this unit. The text also discusses force as a vector and introduces a way to visualize multiple forces acting on an object: the free-body diagram. Notice the free-body diagrams drawn for specific examples in Figures 4.5 and 4.6.

### 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.As you read, pay attention to the example which applies Newton's Third Law of Motion to a swimmer in a pool in Figure 4.9. When the swimmer kicks off the wall of the pool to begin swimming, the swimmer exerts a force toward the wall.

Because of the Third Law, the wall also exerts an opposing force back on the swimmer. The force by the wall on the swimmer is equal in magnitude, but opposite in direction of the force exerted by the swimmer on the wall. In the other axis, gravity exerts a force toward the earth on the swimmer, but interestingly enough, the swimmer is also exerting an equal amount of force on the Earth pulling it up toward them. These are both examples of Newton's Third Law in action. See another example of determining the forces in a given system in Example 4.3.

This lecture accompanies what you just read.

Watch this video for another presentation of Newton's Third Law.

Watch these two videos for examples of action-reaction pairs of forces applied between two objects in contact with each other. They will help you solve problems using Newton's Third Law.

### 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 . 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.As you read, note that we can use Newton's Second Law of Motion to solve problems that involve forces. You should follow the following four steps when solving these types of problems.

1. Sketch the system described in the problem.

2. Identify forces and draw the forces on the sketch.

3. Draw a free-body diagram of the forces acting on the system.

4. Use Newton's Second Law of Motion to solve the problem.As you read this text pay attention to the worked examples of how to solve dynamics problems using the strategies we discussed previously. See examples 4.7, 4.8, and 4.9. We will discuss each of the forces involved soon.

Watch this video from 7:24 to 8:35 for a brief explanation of the force due to gravity that we call weight.

What is the difference between mass and weight? This video will go into the differences of the two concepts and situations where knowing these differences will be useful.

This video demonstrates the characteristics of the forces of tension, friction, weight, and normal when solving problems related to forces.

This video explores gravity, one of the fundamental forces. The narrator explains gravitational interactions in terms of the gravitational field and describes when flat-earth-gravity is a valid approximation.

- This video explores types of forces: normal contact force, tension, friction, air resistance, magnetic force, electrostatic force, and gravitational force. It explains that force is a push or a pull that acts on an object.

Forces are vector quantities because they have both magnitude and direction, and so can be represented by an arrow. Scalar quantities have only magnitude and no direction. When several forces act on an object they can be replaced by a single force that has the same effect. This single force is called the resultant force.

### 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.

Read this text to see an example of how two objects exert gravitational forces on each other in a straight line in Figure 6.21. The force between two objects is directly related to the product of the masses and is inversely proportional to the distance between the objects squared. For two objects with masses and and radius , we can write this as , where is the gravitational constant,

As you can see from the formula, distance plays a large role in the gravitational force acting between two masses. If two masses feel an initial attractive force due to gravity, and then become twice as far from each other, they will now experience a quarter the force as before.Watch these two videos for a demonstration of using the Universal Law of Gravitation for finding the local acceleration on Earth's surface and of a space station near Earth.

Watch this video for detailed analysis of gravity as it applies to astronauts flying high above the atmosphere.

Watch this video to analyze why things fall at the same rate. It uses the Universal Law of Gravitation to prove constant acceleration on Earth for all objects.

### 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.Read this section, which discusses normal force.

Watch this video until 4:37.

This video explains normal force: the force of contact. Again, "normal" essentially means perpendicular. The opposing force is the normal force.

This video introduces the concept of normal force and compares it to the weight of an object.

This video discusses the concept of normal force as it deviates from being exactly equal to the weight of an object, such as inside an accelerating elevator.

These two videos describe more complicated situations dealing with multiple forces (some going diagonal) accompanying normal force.

This video puts the forces we have discussed together, as in the situation of an inclined plane.

### 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.

As you read, pay attention to the example of tension in Figure 4.15 as it talks about how tension is distributed along a rope carrying a weight.

Watch this video from 6:34 to 7:03.

This video explains how tension forces are directed in a simple situation and how it fits in with other classical forces.

### 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:

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 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.Watch this video from 4:37 to 6:34.

The text explains the fundamentals of friction that we discussed earlier, but in more detail.

This video demonstrates the difference between kinetic and static friction.

This lecture accompanies what you just read. It talks about the equations of kinetic and static friction forces, and the concepts behind friction itself. Watch the video until the 6:12 mark, where Clements begins discussing springs and other material we will cover in another Saylor course, PHYS102 Introduction to Electromagnetism.

Watch this video to learn about the differences between static and kinetic friction, and why maximum static friction is generally stronger than kinetic friction.

This video explains how to solve basic friction problems involving static and kinetic cases.

### Unit 4 Assessment

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