### 4.1 Development of Force Concept

- Dynamics is the study of how forces affect the motion of objects.
- Force is a push or pull that can be defined in terms of various standards, and it is a vector having both magnitude and direction.
- External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.

### 4.2 Newton’s First Law of Motion: Inertia

- Newton’s first law of motion states that a body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force. This is also known as the law of inertia.
- Inertia is the tendency of an object to remain at rest or remain in motion. Inertia is related to an object’s mass.
- Mass is the quantity of matter in a substance.

### 4.3 Newton’s Second Law of Motion: Concept of a System

- Acceleration, $\mathbf{\text{a}}$, is defined as a change in velocity, meaning a change in its magnitude or direction, or both.
- An external force is one acting on a system from outside the system, as opposed to internal forces, which act between components within the system.
- Newton’s second law of motion states that the acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass.
- In equation form, Newton’s second law of motion is $\mathbf{\text{a}}=\frac{{\mathbf{\text{F}}}_{\text{net}}}{m}$.
- This is often written in the more familiar form: ${\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}$.
- The weight $\mathbf{\text{w}}$ of an object is defined as the force of gravity acting on an object of mass $m$. The object experiences an acceleration due to gravity $\mathbf{\text{g}}$:
$$\mathbf{\text{w}}=m\mathbf{\text{g}}.$$
- If the only force acting on an object is due to gravity, the object is in free fall.
- Friction is a force that opposes the motion past each other of objects that are touching.

### 4.4 Newton’s Third Law of Motion: Symmetry in Forces

- Newton’s third law of motion represents a basic symmetry in nature. It states: Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts.
- A thrust is a reaction force that pushes a body forward in response to a backward force. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.

### 4.5 Normal, Tension, and Other Examples of Forces

- When objects rest on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force, $\mathbf{\text{N}}$.
- When objects rest on a non-accelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object:
$$N=\text{mg}.$$
- When objects rest on an inclined plane that makes an angle $\theta $ with the horizontal surface, the weight of the object can be resolved into components that act perpendicular (${\mathbf{\text{w}}}_{\perp}$) and parallel (${\mathbf{\text{w}}}_{\parallel}$
**)**to the surface of the plane. These components can be calculated using:$${w}_{\parallel}=w\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}(\theta )=\text{mg}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}(\theta )$$$${w}_{\perp}=w\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}(\theta )=\text{mg}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}(\theta ).$$ - The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension, $\mathbf{\text{T}}$. When a rope supports the weight of an object that is at rest, the tension in the rope is equal to the weight of the object:
$$T=\text{mg}.$$
- In any inertial frame of reference (one that is not accelerated or rotated), Newton’s laws have the simple forms given in this chapter and all forces are real forces having a physical origin.

### 4.6 Problem-Solving Strategies

- To solve problems involving Newton’s laws of motion, follow the procedure described:
- Draw a sketch of the problem.
- Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
- Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the $x$-direction) then ${F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=0$. If the object does accelerate in that direction, ${F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=\text{ma}$.
- Check your answer. Is the answer reasonable? Are the units correct?

### 4.7 Further Applications of Newton’s Laws of Motion

- Newton’s laws of motion can be applied in numerous situations to solve problems of motion.
- Some problems will contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether ${F}_{\text{net}}=\text{ma}$ or ${F}_{\text{net}}=0$ .
- The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating, the normal force will be less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force will always be less than the full weight of the object.
- Some problems will contain various physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics in order to solve these problems of motion.

### 4.8 Extended Topic: The Four Basic Forces—An Introduction

- The various types of forces that are categorized for use in many applications are all manifestations of the
*four basic forces*in nature. - The properties of these forces are summarized in Table 4.1.
- Everything we experience directly without sensitive instruments is due to either electromagnetic forces or gravitational forces. The nuclear forces are responsible for the submicroscopic structure of matter, but they are not directly sensed because of their short ranges. Attempts are being made to show all four forces are different manifestations of a single unified force.
- A force field surrounds an object creating a force and is the carrier of that force.