Physics

Chapter 5 Newton’s Laws of Motion

Imron Rosyadi

Introduction to Dynamics

Learning Objectives

By the end of this section, you will be able to:

• Distinguish between kinematics and dynamics.
• Understand the definition of force.
• Identify simple free-body diagrams.
• Define the SI unit of force, the newton.
• Describe force as a vector.

Welcome to Chapter 5: Newton’s Laws of Motion. In this chapter, we transition from describing motion, which is kinematics, to understanding the causes of motion, known as dynamics. We’ll explore the fundamental concepts of force, its measurement, and how it’s represented as a vector. We’ll also introduce the critical tool of free-body diagrams, which will be essential for analyzing forces.

Kinematics vs. Dynamics

Kinematics

• Describes how objects move.
• Focuses on velocity and acceleration.
• Doesn’t consider the causes of motion.

Dynamics

• Studies why objects move.
• Investigates forces that cause changes in motion.
• Built upon Isaac Newton’s laws of motion.

Kinematics, which we’ve already covered, tells us about the how of motion – things like position, velocity, and acceleration. Dynamics, on the other hand, delves into the why. It’s about the forces that act on objects and systems, causing them to accelerate or change their state of motion. Newton’s laws are the cornerstone of dynamics, providing a universal framework for understanding motion both on Earth and in space.

Isaac Newton’s Contribution

Figure 5.2 Isaac Newton (1642–1727) published his monumental work, Philosophiae Naturalis Principia Mathematica, in 1687.

Isaac Newton’s work in the 17th century revolutionized our understanding of physics. His laws of motion, along with his law of gravity and the invention of calculus, laid the foundation for classical mechanics. These laws are incredibly broad and simple, applying across vast scales, from everyday objects to celestial bodies. It’s important to note that Newtonian mechanics is valid for objects moving at speeds much less than the speed of light and for objects larger than molecules. For extreme conditions, modern physics, including relativity and quantum mechanics, provides a more accurate description.

What is a Force?

• A force is a push or a pull on an object.
• Forces have both magnitude and direction, making them vector quantities.
• Represented by arrows: length indicates magnitude, arrowhead indicates direction.

Figure 5.3 (a) Two ice skaters pushing a third.

• Forces add like other vectors.
• Resultant force is the vector sum of all individual forces.
• Head-to-tail or trigonometric methods can be used for addition.

At its most basic, a force is a push or a pull. This intuitive definition is actually quite powerful. Since forces have both a strength (magnitude) and a direction, they are vector quantities. This means they combine according to vector addition rules. Just like two people pushing on someone from different angles, the total effect, or the resultant force, is determined by summing these individual force vectors.

Free-Body Diagrams (FBDs)

• A sketch showing all external forces acting on an object or system.
• The object is represented as a single isolated point.
• Forces are shown as vectors originating from this point.

Why are FBDs important?

• Help analyze forces.
• Crucial for applying Newton’s laws.
• Only external forces affect the motion of the system.

Figure 5.4 Sample free-body diagrams.

Free-body diagrams are perhaps the most important tool you’ll learn in this chapter for problem-solving. They help us simplify complex situations by isolating the object of interest and drawing only the forces acting on it from its surroundings. We represent the object as a point to focus solely on the forces. Importantly, we only include external forces, as internal forces within the system cancel out and don’t affect the system’s overall motion.

Types of Forces

1. Contact Forces

• Result from direct physical contact.
• Examples:
  • Push on a door.
  • Friction between tires and road.
  • Normal force from a surface.

2. Field Forces

• Act without physical contact.
• Depend on a “field” in space.
• Examples:
  • Gravitational force (weight)
  • Electromagnetic force
  • Strong nuclear force
  • Weak nuclear force

Forces can be broadly categorized into two types: contact forces and field forces. Contact forces are exactly what they sound like – they occur when objects are physically touching. Pushing a cart, friction, or a table supporting a book are all examples. Field forces, on the other hand, don’t require contact. Gravity is the most common example in classical mechanics; Earth exerts a gravitational force on you even when you’re jumping in the air. Electromagnetic forces are also field forces, responsible for static electricity, magnetism, and even the atomic interactions that give rise to contact forces at a microscopic level. The strong and weak nuclear forces operate at subatomic scales and aren’t relevant in Newtonian mechanics.

Force Units and Vector Notation

• SI Unit of Force: Newton (N)
  • Defined as: \(1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2\)
  • Roughly the weight of a small apple.
• Vector Notation:
  • 2D force: \(\vec{F} = a\hat{i} + b\hat{j}\)
  • 3D force: \(\vec{F} = a\hat{i} + b\hat{j} + c\hat{k}\)
• Net External Force (\(\vec{F}_{\text{net}}\) or \(\sum \vec{F}\)):
  • The vector sum of all external forces acting on an object.
  • \(\vec{F}_{\text{net}} = \sum \vec{F} = \vec{F}_1 + \vec{F}_2 + \dots\)

The standard unit for force in the International System of Units is the Newton, named after Isaac Newton. One Newton is defined as the force required to accelerate a 1-kilogram mass at a rate of 1 meter per second squared. To give you a feel for it, a small apple weighs about 1 Newton. Since force is a vector, we use vector notation to describe its direction. For 2D problems, we use \(\hat{i}\) and \(\hat{j}\) for the x and y directions, and for 3D, we add \(\hat{k}\) for the z-direction. The most important concept here is the net external force. This is the overall, resultant force acting on an object, found by vectorially summing all the individual external forces. This net force is what determines an object’s acceleration.

Example: Resultant Force

• Skater 1 pushes with \(\vec{F}_1 = 30.0\hat{i}\text{ N}\) (to the right).
• Skater 2 pushes with \(\vec{F}_2 = 40.0\hat{j}\text{ N}\) (vertically upwards).

Find the net force (\(\vec{F}_{\text{net}}\)) on the third skater.

\(\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2\)
\(\vec{F}_{\text{net}} = (30.0\hat{i} + 40.0\hat{j})\text{ N}\)

Magnitude:
\(F_{\text{net}} = \sqrt{(30.0\text{ N})^2 + (40.0\text{ N})^2} = 50.0\text{ N}\)

Direction:
\(\theta = \tan^{-1}\left(\frac{40.0}{30.0}\right) = 53.1^\circ\) from the positive x-axis.

Figure 5.3 (b) Free-body diagram.

Let’s apply these concepts to the ice skaters. If one skater pushes right with 30 Newtons and another pushes forward with 40 Newtons, how do we find the total force? Since forces are vectors, we add them vectorially. The net force is the sum of these two perpendicular forces. Using the Pythagorean theorem for the magnitude and the inverse tangent for the angle, we find a resultant force of 50 Newtons at an angle of 53.1 degrees from the horizontal. This example demonstrates how to combine forces using vector addition.

Newton’s First Law of Motion

Learning Objectives

By the end of this section, you will be able to:

• Describe Newton’s first law of motion.
• Recognize friction as an external force.
• Define inertia.
• Identify inertial reference frames.
• Calculate equilibrium for a system.

Now we dive into Newton’s first law, often called the law of inertia. This law describes what happens to an object when there is no net force acting on it. It introduces fundamental concepts like inertia and equilibrium, and also the idea of an inertial reference frame, which is crucial for applying Newton’s laws correctly.

Newton’s First Law: The Law of Inertia

A body at rest remains at rest or, if in motion, remains in motion at constant velocity unless acted on by a net external force.

• Constant velocity means constant speed and constant direction.
• This law explains that a net external force is required for any change in velocity (acceleration).

Figure 5.7 A hockey puck illustrates Newton’s first law.

Key ideas:

• Status quo: Objects resist changes to their current state of motion.
• Cause and effect: Any change in motion must have an external cause (a net force).

Newton’s first law is deceptively simple, yet profoundly important. It states that objects tend to maintain their state of motion. If something is at rest, it stays at rest. If it’s moving, it keeps moving at the same speed in the same direction. The crucial part is “unless acted on by a net external force.” This means that if an object’s velocity changes – whether its speed or its direction – there must be an external force responsible for that change. Friction is a common external force that slows objects down in our everyday experience. Without friction, an object would continue moving indefinitely.

Inertia and Mass

• Inertia: The ability of an object to resist changes in its motion (resist acceleration).
• Mass: A quantitative measure of inertia.

• More mass \(\implies\) more inertia.
• Harder to change the motion of a massive object.

Figure 5.11 Same force, different masses, different accelerations.

Inertia is a fundamental property of matter. It’s the intrinsic resistance an object has to any change in its state of motion. Think about pushing a basketball versus pushing a boulder – the boulder is much harder to get moving or stop, because it has more inertia. The quantitative measure of this inertia is called mass. The more mass an object has, the more inertia it possesses, and the more force is required to change its velocity.

Inertial Reference Frames

• A reference frame where Newton’s first law is valid.
• An object’s velocity relative to this frame is constant if the net force on it is zero.

Important

A reference frame moving at constant velocity relative to an inertial frame is also inertial. A reference frame accelerating relative to an inertial frame is not inertial.

• For most problems, a reference frame fixed on Earth is a sufficiently accurate approximation of an inertial frame.

Newton’s first law also helps us define a special type of reference frame: an inertial reference frame. In such a frame, if there are no net external forces acting on an object, it will either remain at rest or move with constant velocity. The key here is constant velocity. If a frame is accelerating, then an object with no net force on it would appear to accelerate, violating the first law. For practical purposes in this course, we’ll generally treat Earth as an inertial frame, even though it technically accelerates due to its rotation and orbit. For most everyday problems, this approximation is perfectly acceptable.

Equilibrium

• A system is in equilibrium when the forces on it are balanced.
• The net external force is zero: \(\vec{F}_{\text{net}} = \vec{0}\).
• Consequently, the velocity is constant: \(\vec{v} = \text{constant}\).
  • This includes the case where \(\vec{v} = \vec{0}\) (at rest).

Figure 5.9 A car (a) parked and (b) moving at constant velocity.

Two types of equilibrium:

1. Static Equilibrium: Object is at rest (\(\vec{v} = \vec{0}\)).
2. Dynamic Equilibrium: Object is moving at a constant velocity (\(\vec{v} = \text{constant and non-zero}\)).

Equilibrium is a state where the net external force on an object is zero. According to Newton’s first law, this means the object’s velocity will be constant. This constant velocity can be zero, in which case we call it static equilibrium – like a parked car. Or, the constant velocity can be non-zero, meaning the object is moving at a steady speed in a straight line – this is dynamic equilibrium, like a car cruising on the highway at a constant speed. In both cases, the forces are balanced.

Newton’s Second Law of Motion

Learning Objectives

By the end of this section, you will be able to:

• Distinguish between external and internal forces.
• Describe Newton’s second law of motion.
• Explain the dependence of acceleration on net force and mass.

Having understood what happens when forces are balanced, we now turn to Newton’s second law, which describes what happens when forces are unbalanced. This law provides the quantitative relationship between force, mass, and acceleration, allowing us to predict how objects will move when subjected to external forces.

Net External Force and Acceleration

• A net external force causes a change in motion (acceleration).
• External forces: Act on the system from outside.
• Internal forces: Act between elements within the system.
  • Internal forces cancel out and do not affect the system’s overall motion.

• Acceleration (\(\vec{a}\)) is directly proportional to the net external force (\(\vec{F}_{\text{net}}\)).
• Acceleration (\(\vec{a}\)) is inversely proportional to the system’s mass (\(m\)).

Figure 5.10 Different forces on the same mass produce different accelerations.

Newton’s second law connects the concepts of force, mass, and acceleration. First, it clarifies that only a net external force can cause an object or system to accelerate. Internal forces within a system, like the force a driver exerts on the steering wheel, don’t affect the car’s overall motion because they cancel out within the system. The law then quantifies this relationship: acceleration is directly proportional to the net external force – a bigger net force means bigger acceleration. And, acceleration is inversely proportional to the mass – a more massive object will have less acceleration for the same applied force.

Newton’s Second Law: The Equation

The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system and is inversely proportional to its mass.

• Vector Form: \[ \vec{a} = \frac{\vec{F}_{\text{net}}}{m} \quad \text{or} \quad \vec{F}_{\text{net}} = m\vec{a} \]
• Scalar Form (for magnitudes): \[ F_{\text{net}} = ma \]

Note

Newton’s second law is a cause-and-effect relationship: force causes acceleration. Its validity is confirmed by experimental observation.

This is the mathematical heart of dynamics: Newton’s second law. It states that the acceleration of an object is determined by the net external force acting on it, divided by its mass. The acceleration vector will always point in the same direction as the net force vector. Remember that this equation represents a fundamental law of nature, discovered through observation and verified by countless experiments. When solving problems, we often use the scalar form for magnitudes, but always remember the vector nature of force and acceleration. The free-body diagram you draw is the essential starting point for applying this law to a problem.

Components of Newton’s Second Law

The vector equation \(\vec{F}_{\text{net}} = m\vec{a}\) can be broken down into three independent component equations:

\[ \sum F_x = ma_x \]
\[ \sum F_y = ma_y \]
\[ \sum F_z = ma_z \]

• This allows us to analyze motion in each dimension separately.
• The net force in a specific direction causes acceleration only in that direction.

Since force and acceleration are vectors, Newton’s second law can be expressed in terms of components. This is incredibly useful for solving problems, especially in two or three dimensions. We can break down all forces into their x, y, and z components, and then apply Newton’s second law independently to each direction. The sum of forces in the x-direction will equal mass times the x-component of acceleration, and similarly for the y and z directions. This highlights that forces in one dimension do not affect motion in a perpendicular dimension.

Newton’s Second Law and Momentum

• Newton’s original statement of the second law was in terms of momentum.
• Momentum (\(\vec{p}\)): The product of an object’s mass (\(m\)) and its velocity (\(\vec{v}\)). \[ \vec{p} = m\vec{v} \]
• Momentum Form of Newton’s Second Law: \[ \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} \] (The instantaneous rate at which a body’s momentum changes is equal to the net force acting on the body.)

Note

When mass (\(m\)) is constant, this reduces to \(\vec{F}_{\text{net}} = m\frac{d\vec{v}}{dt} = m\vec{a}\).

It’s interesting to note that Newton originally formulated his second law in terms of momentum. Momentum, defined as mass times velocity, is a measure of the “quantity of motion” an object possesses. Since velocity is a vector, momentum is also a vector. Newton’s second law, in its momentum form, states that the net force on an object is equal to the rate of change of its momentum. For most problems in this chapter, where mass is constant, this definition simplifies to the more familiar \(\vec{F}_{\text{net}} = m\vec{a}\), because the derivative of mass times velocity with respect to time becomes mass times the derivative of velocity, which is acceleration.

Mass and Weight

Learning Objectives

By the end of this section, you will be able to:

• Explain the difference between mass and weight.
• Explain why objects falling through the air are never truly in free fall.
• Describe the concept of weightlessness.

In everyday language, mass and weight are often used interchangeably, but in physics, they are distinct and important concepts. This section clarifies this difference and introduces the gravitational force, which is essentially what we call weight. We’ll also touch upon the practical implications of air resistance and the idea of “weightlessness.”

Mass vs. Weight

Mass (\(m\))

• An intrinsic property of an object.
• A measure of the amount of matter in an object.
• A measure of an object’s inertia.
• Does not vary with location.
• SI Unit: kilogram (kg).

Weight (\(\vec{w}\))

• The gravitational force on an object.
• Pull of a large body (e.g., Earth) on an object.
• A force vector, always directed downwards.
• Varies with location (depends on local gravity, \(g\)).
• SI Unit: newton (N).

This slide highlights the crucial distinction between mass and weight. Mass is an intrinsic property of an object – it’s how much “stuff” is in it and how much it resists acceleration. Your mass is the same whether you’re on Earth, the Moon, or in deep space. Weight, however, is a force – specifically, the force of gravity acting on your mass. Since the acceleration due to gravity (\(g\)) changes depending on your location (e.g., stronger on Earth than on the Moon), your weight will also change. It’s a common mistake to use them interchangeably, but remember, mass is a scalar quantity (amount of matter/inertia), and weight is a vector quantity (gravitational force).

Calculating Weight

• Weight (\(\vec{w}\)) is the gravitational force acting on an object of mass (\(m\)).

• It is a specific application of Newton’s second law (\(\vec{F}_{\text{net}} = m\vec{a}\)).

• Vector Form: \[ \vec{w} = m\vec{g} \]

• Scalar Form (magnitude): \[ w = mg \]

Note

On Earth, \(g \approx 9.80 \text{ m/s}^2\). So, a 1.00-kg object on Earth weighs \(w = (1.00 \text{ kg})(9.80 \text{ m/s}^2) = 9.80 \text{ N}\).

The relationship between weight and mass is simple: weight is equal to mass multiplied by the acceleration due to gravity (\(g\)). On Earth’s surface, \(g\) is approximately 9.80 m/s², so a 1-kilogram object weighs about 9.8 Newtons. It’s important to remember that ‘g’ is not constant everywhere; it varies slightly across Earth’s surface and significantly on other celestial bodies like the Moon. Therefore, an object’s weight is dependent on its location.

Free Fall and Weightlessness

• Free Fall: The state where the only force acting on an object is gravity.
  • Objects in free fall accelerate at \(g\).
  • Objects falling through air are not truly in free fall due to air resistance.
• “Weightlessness” (popular media definition):
  • Refers to the sensation of being in free fall.
  • Often experienced in orbit, where objects are continuously falling around Earth.
  • In physics, “weightlessness” means zero gravitational force, which is different from free fall.

Warning

Do not confuse the everyday use of “weightlessness” with its precise physics definition.

When we talk about free fall in physics, it means an object is experiencing only the force of gravity. In reality, on Earth, objects falling through the air are always subject to air resistance, so they’re not truly in free fall. The term “weightlessness” is often used in popular media to describe the feeling astronauts experience in orbit. However, in physics, this sensation is more accurately described as being in continuous free fall. True weightlessness, meaning zero gravitational force, would only occur far away from any massive body in space. It’s crucial to understand this distinction for clarity in physics.

Newton’s Third Law

Learning Objectives

By the end of this section, you will be able to:

• State Newton’s third law of motion.
• Identify the action and reaction forces in different situations.
• Apply Newton’s third law to define systems and solve problems of motion.

Newton’s third law introduces a fundamental symmetry in nature: forces always come in pairs. This law, often summarized as “action-reaction,” is crucial for understanding how objects interact and for correctly identifying forces when solving problems. We’ll learn how to identify these force pairs and apply the law to various scenarios, emphasizing that these forces act on different objects.

Newton’s Third Law: Action-Reaction

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 it exerts.

• Mathematically: \(\vec{F}_{\text{AB}} = -\vec{F}_{\text{BA}}\)
  • \(\vec{F}_{\text{AB}}\) is the force exerted by A on B.
  • \(\vec{F}_{\text{BA}}\) is the force exerted by B on A.

Figure 5.16 Swimmer pushing off a wall.

Key characteristics of action-reaction pairs:

• Always equal in magnitude.
• Always opposite in direction.
• Always act on different bodies (systems).
  • Therefore, they do not cancel each other out within a single system.

Newton’s third law is perhaps the most misunderstood of his three laws. It states that for every action, there is an equal and opposite reaction. This means forces always come in pairs. The crucial point, often missed, is that these action-reaction forces always act on different objects. For example, when a swimmer pushes against a wall (action force on the wall), the wall pushes back on the swimmer (reaction force on the swimmer). The swimmer accelerates due to the force from the wall, not the force she exerted on the wall. Because these forces act on different systems, they never cancel each other out when analyzing the motion of a single object.

Examples of Action-Reaction Pairs

• Walking: Your feet push backward on the ground; the ground pushes forward on you.
• Car Acceleration: Drive wheels push backward on the ground; the ground pushes forward on the wheels.
• Rockets: Expel gas backward; gas exerts a forward thrust on the rocket.
• Helicopters/Birds: Push air down; air exerts an upward lift force.

Figure 5.18 Runner pushing on the ground.

Figure 5.17 Climber pulling on a rope.

Newton’s third law is constantly at play around us. When you walk, your foot pushes backward on the ground, and in response, the ground pushes forward on your foot, propelling you forward. Similarly, rockets don’t “push off” the air; they push exhaust gases backward, and the gases, in turn, push the rocket forward. This principle of action and reaction is fundamental to all forms of locomotion and force transmission. Remember, always identify the two interacting objects to correctly identify an action-reaction pair.

Choosing the System

• The choice of the “system of interest” is critical when applying Newton’s laws.
• Only external forces (forces from outside the chosen system) cause the system to accelerate.
• Internal forces (forces between components within the system) cancel out within that system and do not contribute to its net acceleration.

Tip

When drawing a free-body diagram, always be clear about what constitutes your “system” to correctly identify external forces.

One of the most important skills in dynamics is defining your “system of interest.” This decision dictates which forces are considered external and which are internal. Internal forces, even though they exist (like the professor pushing the cart), will cancel out if both the professor and the cart are part of your system. Only forces acting on the system from the outside will contribute to its net acceleration. Carefully choosing your system simplifies problem-solving by allowing you to disregard internal forces.

Common Forces

Learning Objectives

By the end of this section, you will be able to:

• Define normal and tension forces.
• Distinguish between real and fictitious forces.
• Apply Newton’s laws of motion to solve problems involving a variety of forces.

In this section, we’ll categorize and define several common types of forces that you’ll encounter frequently in physics problems. Understanding these forces – normal force, tension, friction, and spring force – is essential for constructing accurate free-body diagrams and applying Newton’s laws effectively. We’ll also briefly distinguish between real and fictitious forces.

Normal Force (\(\vec{N}\))

• A support force exerted by a surface on an object in contact with it.
• Always acts perpendicular (“normal”) to the surface of contact.

• Counteracts the component of weight perpendicular to the surface.

On a horizontal surface (object at rest):
\(N = w = mg\)
(magnitude)

Figure 5.21 (b) Normal force supporting a bag of dog food.

The normal force is a contact force that prevents objects from passing through surfaces. It’s always perpendicular to the surface of contact. For an object resting on a horizontal surface, the normal force will be equal in magnitude and opposite in direction to the object’s weight. However, as we’ll see with inclined planes, the normal force isn’t always equal to the full weight; it only balances the component of weight that is perpendicular to the surface.

Normal Force on an Incline

• When an object is on an inclined plane, gravity (\(\vec{w}\)) is resolved into two components:
  • Perpendicular component (\(w_y\)): Perpendicular to the surface, balanced by the normal force.
  • Parallel component (\(w_x\)): Parallel to the surface, causing motion down the incline (if no opposing forces).

Figure 5.22 Skier on a slope.

Components of Weight:
\(w_x = w \sin\theta = mg \sin\theta\)
\(w_y = w \cos\theta = mg \cos\theta\)

Normal Force on an Incline:
\(N = mg \cos\theta\)

On an inclined plane, things get a bit more interesting for the normal force. It’s still perpendicular to the surface, but now the object’s weight is no longer entirely perpendicular to the surface. We resolve the weight vector into two components: one perpendicular to the incline (\(w_y\)) and one parallel to the incline (\(w_x\)). The normal force will then balance only the perpendicular component of the weight, meaning \(N = mg \cos\theta\). The parallel component, \(mg \sin\theta\), is what tends to make the object slide down the incline. This is a common setup in physics problems, so it’s good to visualize these components.

Tension Force (\(\vec{T}\))

• A pulling force transmitted along the length of a flexible connector (rope, cable, string, tendon, etc.).
• Always acts parallel to the length of the connector.
• “You can’t push a rope.”

Object hanging at rest:
\(T = w = mg\)
(magnitude, neglecting rope mass)

• Tension is uniform throughout a massless, frictionless rope.

Figure 5.24 Mass hanging from a rope.

Tension is another common contact force, specifically a pulling force transmitted through flexible connectors like ropes or cables. It always acts along the length of the connector. A key concept is that for an ideal (massless and frictionless) rope, the tension is the same throughout its length. If an object is hanging motionless from a rope, the tension in the rope is equal to the object’s weight, balancing the gravitational force.

Real vs. Fictitious Forces

Real Forces

• Have a physical origin (e.g., gravity, electromagnetism).
• Exist in all reference frames.
• Always have an identifiable reaction force (Newton’s 3rd Law).

Fictitious Forces

• Arise when an observer is in an accelerating (non-inertial) reference frame.
• Are not real physical forces; they are apparent forces.
• No identifiable reaction force.
• Examples: Coriolis effect (in rotating frames), apparent “push” when a car brakes.

This distinction is important for a deeper understanding of forces. Real forces, like gravity or the push from a wall, have a fundamental physical origin and always adhere to Newton’s third law. Fictitious forces, on the other hand, are not actual interactions but rather appear to exist because the observer is in a non-inertial, accelerating reference frame. A classic example is the feeling of being pushed forward when a car brakes abruptly – there’s no “force” pushing you forward, but your inertia makes it feel that way relative to the decelerating car. We typically work in inertial frames where only real forces are considered.

Other Common Forces

• Friction (\(\vec{f}\)):
  • A resistive force that opposes motion or the tendency of motion.
  • Acts parallel to the contact surface.
  • Will be discussed in detail in the next chapter.
• Spring Force (\(\vec{F}_{\text{spring}}\)):
  • Exerted by a deformed spring (stretched or compressed).
  • Hooke’s Law: \(\vec{F}_{\text{spring}} = -k\Delta\vec{x}\)
    • \(k\): spring constant (stiffness).
    • \(\Delta\vec{x}\): displacement from relaxed position.
  • Acts to restore the spring to its relaxed shape.

Beyond normal force and tension, two other common forces you’ll encounter are friction and spring force. Friction is a resistive force that always opposes motion or the tendency of motion between surfaces in contact. We’ll dedicate more time to friction in the next chapter. Spring force is unique to elastic objects like springs. It’s a restoring force that pushes or pulls the spring back to its original, relaxed position. Hooke’s Law describes this force, stating it’s proportional to the displacement from the relaxed state and acts in the opposite direction of that displacement.

Drawing Free-Body Diagrams (Recap)

Learning Objectives

By the end of this section, you will be able to:

• Explain the rules for drawing a free-body diagram.
• Construct free-body diagrams for different situations.

We revisit free-body diagrams to solidify your understanding of this essential problem-solving tool. Mastering the construction of FBDs is foundational to successfully applying Newton’s laws. We’ll reinforce the rules and look at more complex scenarios to build your confidence.

Problem-Solving Strategy: Constructing FBDs

1. Isolate and Represent:
   • Draw the object of interest as a point (if treating as a particle) or a simple sketch.
   • Place it at the origin of an xy-coordinate system.
2. Identify ALL External Forces:
   • Include all forces acting ON the object (e.g., weight, normal, tension, friction, applied forces).
   • Represent forces as vectors originating from the object.
   • DO NOT include internal forces or forces the object exerts on other objects.
   • DO NOT include the net force (\(\vec{F}_{\text{net}}\)).
3. Resolve Components (Optional but Recommended):
   • Break down inclined forces into x- and y-components (especially useful if axes are aligned with an incline).
   • Draw a squiggly line through the original vector to show it’s replaced by its components.
4. Multiple Objects = Multiple FBDs:
   • If a problem involves several interacting objects, draw a separate FBD for each one.

Tip

Indicate acceleration (\(\vec{a}\)) outside the FBD to remember the direction of net force.

Let’s review the step-by-step process for constructing effective free-body diagrams. First, clearly define and isolate your object of interest, representing it simply, often as a point. Second, identify all the external forces acting on that object. This means forces from its surroundings, like gravity, surface contact forces, or tension from ropes. Remember, internal forces or forces the object exerts on others are excluded. You also don’t draw the net force itself; the FBD helps you find the net force. Third, if forces are at angles, it’s often helpful to resolve them into their x and y components, aligning your axes to simplify calculations. Finally, for problems with multiple interacting objects, draw a separate FBD for each object. A helpful tip is to draw the acceleration vector nearby but distinct from the FBD to remind you of the object’s overall motion.

Example FBD: Sled Pulled at an Angle

Figure 5.31 (a) Sled pulled by force P.

Analysis:

• Object: Sled
• Forces:
  1. Weight (\(\vec{w}\)): Downward (due to Earth’s gravity).
  2. Normal Force (\(\vec{N}\)): Upward (from the ground, perpendicular to surface).
  3. Applied Force (\(\vec{P}\)): At 30° angle (the pull).
  4. Friction (\(\vec{f}\)): Opposing motion (if moving or tending to move).

Let’s walk through an example. Imagine a sled being pulled across the snow by a rope at an angle. To draw its FBD, first, we represent the sled as a point. Then, we identify the forces acting on it. There’s its weight pulling down, the normal force from the snow pushing up, the tension from the rope pulling it at an angle, and friction from the snow opposing its motion. The diagram shows how these forces are represented as vectors from the central point. If we then decide to resolve the angled pulling force into horizontal and vertical components, we would indicate that the original vector has been replaced by its components.

Key Takeaways

• Dynamics studies the forces that cause motion, building on Newton’s Laws.
• A force is a push or pull, a vector quantity measured in Newtons (N).
• Free-body diagrams (FBDs) are essential tools that isolate an object and show all external forces acting on it.
• Newton’s First Law (Law of Inertia): An object’s velocity remains constant unless acted upon by a net external force.
  • Inertia is an object’s resistance to changes in motion, measured by its mass.
  • Equilibrium occurs when the net force is zero (constant velocity or at rest).
• Newton’s Second Law: \(\vec{F}_{\text{net}} = m\vec{a}\). The net external force causes an object to accelerate in the same direction, inversely proportional to its mass.
  • This vector equation can be resolved into components (\(\sum F_x = ma_x\), etc.).
  • It relates to momentum: \(\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}\).

Key Takeaways

• Mass is an intrinsic property (amount of matter/inertia); weight is the gravitational force (\(w=mg\)), which varies with location.
• Newton’s Third Law (Action-Reaction): Forces always occur in pairs; \(\vec{F}_{\text{AB}} = -\vec{F}_{\text{BA}}\). These forces are equal in magnitude, opposite in direction, and act on different objects.
• Common Forces:
  • Normal Force (\(\vec{N}\)): Perpendicular to a contact surface.
  • Tension (\(\vec{T}\)): Pulling force along a flexible connector.
  • Friction (\(\vec{f}\)): Resists motion.
  • Spring Force (\(\vec{F}_{\text{spring}}\)): Restoring force (Hooke’s Law: \(\vec{F}_{\text{spring}} = -k\Delta\vec{x}\)).
• Real forces have physical origins; fictitious forces arise from accelerating reference frames.

Key Equations

Equation	Description
\(\vec{F}_{\text{net}} = \sum \vec{F}\)	Net external force is the vector sum of all external forces.
\(\vec{F}_{\text{net}} = m\vec{a}\)	Newton’s Second Law: Relates net force, mass, and acceleration.
\(\sum F_x = ma_x\)	Component form of Newton’s Second Law for the x-direction.
\(\sum F_y = ma_y\)	Component form of Newton’s Second Law for the y-direction.
\(\sum F_z = ma_z\)	Component form of Newton’s Second Law for the z-direction.
\(\vec{p} = m\vec{v}\)	Definition of momentum.
\(\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}\)	Newton’s Second Law in terms of momentum.
\(\vec{w} = m\vec{g}\)	Weight: gravitational force on an object of mass \(m\).
\(N = mg \cos\theta\)	Normal force on an object on an inclined plane at angle \(\theta\) to the horizontal.
\(\vec{F}_{\text{AB}} = -\vec{F}_{\text{BA}}\)	Newton’s Third Law: Action-reaction pair (forces are equal and opposite, acting on different bodies).
\(\vec{F}_{\text{spring}} = -k\Delta\vec{x}\)	Hooke’s Law: Spring force is proportional to displacement \(\Delta\vec{x}\) from equilibrium, constant \(k\).

Key Terms

Term	Definition
Dynamics	The study of how forces affect the motion of objects and systems.
Force	A push or pull on an object with a specific magnitude and direction. It is a vector quantity.
Newton (N)	The SI unit of force, defined as the force needed to accelerate an object with a mass of 1 kg at a rate of 1 m/s\(^2\).
Free-body Diagram	A sketch showing all external forces acting on an object or system, represented as a single isolated point.
External Force	A force acting on an object or system that originates outside of the object or system.
Internal Force	A force that acts between elements of the system of interest.
Inertia	The property of an object to resist changes in its state of motion (i.e., to resist acceleration).
Mass	A quantitative measure of inertia; the amount of matter in an object.
Inertial Reference Frame	A reference frame in which Newton’s first law is valid (i.e., an object with zero net force moves at constant velocity).
Equilibrium	The state in which the net external force on a system is zero, resulting in constant velocity (which can be zero).
Weight (\(\vec{w}\))	The gravitational force on an object from the nearest large body (e.g., Earth); a vector quantity directed toward the center of the gravitating body.
Normal Force (\(\vec{N}\))	The force exerted by a surface on an object in contact with it that is perpendicular to the surface.
Tension (\(\vec{T}\))	A pulling force that acts along the length of a stretched flexible connector, such as a rope or cable.
Friction (\(\vec{f}\))	A resistive force that opposes the motion or tendency of motion between surfaces in contact.
Spring Force (\(\vec{F}_{\text{spring}}\))	The restoring force exerted by a deformed spring, proportional to the displacement from its relaxed position and acting in the opposite direction (Hooke’s Law).
Real Forces	Forces that have a physical origin and exist independently of the observer’s frame of reference.
Fictitious Forces	Apparent forces that arise when an observer is in an accelerating (non-inertial) reference frame; they do not have a physical origin.