Chapter 04

Newton's Laws of Motion

Introduction to Dynamics

Dynamics is the branch of physics that studies the relationship between motion and the forces that cause it. While kinematics describes how objects move, dynamics explains why they move. The foundational principles of dynamics were first clearly stated by Sir Isaac Newton and are known as Newton's laws of motion.

• Kinematics describes motion in terms of displacement, velocity, and acceleration.

• Dynamics relates motion to its causes, specifically forces.

• Newton's laws were deduced from experimental observations and are fundamental to classical mechanics.

Forces in Physics

Properties of a Force

A force is a physical quantity that can cause an object to accelerate, change direction, or deform. Forces are central to understanding motion and interactions in physics.

• Definition: A force is a push or a pull exerted on an object.

• Interaction: Forces arise from interactions between two objects or between an object and its environment.

• Vector Quantity: Forces have both magnitude and direction, and are represented as vectors.

• Notation: The force vector is commonly denoted as $\vec{F}$.

• Example: Pushing a box to the right applies a force in the direction of the push; pulling a box applies a force in the direction of the pull.

Types of Forces

Normal Force

The normal force is a contact force exerted by a surface perpendicular to the object resting or pressing on it.

• Definition: When an object rests or pushes on a surface, the surface exerts a push on it that is directed perpendicular to the surface.

• Notation: $\vec{n}$

• Contact Force: The normal force only exists when there is contact between the object and the surface.

• Example: A book resting on a table experiences an upward normal force from the table.

Friction Force

The friction force is a contact force that opposes the relative motion or attempted motion between two surfaces in contact.

• Definition: In addition to the normal force, a surface may exert a friction force on an object, directed parallel to the surface.

• Notation: $\vec{f}$

• Contact Force: Friction only acts when two surfaces are in contact.

• Example: Sliding a box across the floor involves a friction force opposing the motion.

Tension Force

The tension force is a pulling force transmitted through a string, rope, cable, or similar object.

• Definition: Tension is the force exerted by a rope or cord when it is pulled tight by forces acting from opposite ends.

• Contact Force: Tension acts along the length of the rope and is directed away from the object.

• Example: A weight hanging from a rope experiences an upward tension force.

• Additional info: Tension is always directed along the rope, away from the object.

Weight (Gravitational Force)

The weight of an object is the force of gravity acting on it. Unlike the previous forces, weight is a long-range force and does not require contact.

• Definition: The pull of gravity on an object is a long-range force that acts over a distance.

• Formula: $w = m g$

• Variables: $m$ is the mass of the object, $g$ is the acceleration due to gravity (approximately $9.8\,\text{m/s}^2$ on Earth).

• Example: A rock falling towards the ground is pulled by the gravitational force.

Units of Force

SI Unit: Newton

The standard unit of force in the International System (SI) is the newton (N).

• Definition: $1\,\text{N} = 1\,\text{kg} \cdot \text{m/s}^2$

• Other Units: In the British system, force is measured in pounds (lb); in the cgs system, force is measured in dynes.

System	Unit of Force	Unit of Mass	Unit of Distance
SI	Newton (N)	kilogram (kg)	meter (m)
British	Pound (lb)	slug	foot (ft)
cgs	Dyne	gram (g)	centimeter (cm)

Vector Nature of Forces

Force Vectors and Superposition

Forces are vectors and can be represented graphically by arrows. The length of the arrow indicates the magnitude, and the direction shows the direction of the force.

• Superposition Principle: Multiple forces acting on an object combine by vector addition to produce a single resultant (net) force.

• Notation: $\vec{R} = \vec{F}_1 + \vec{F}_2 + \cdots + \vec{F}_n$

• Component Vectors: Forces can be decomposed into perpendicular components, typically along the x- and y-axes.

• Example: A force at an angle can be split into horizontal and vertical components using trigonometry.

Newton's First Law of Motion

Law of Inertia

Newton's first law states that an object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Equilibrium: An object is in equilibrium if the net force acting on it is zero.

• Mathematical Statement: $\sum \vec{F} = 0 \implies \vec{a} = 0$

• Inertial Frames: Newton's first law is valid only in inertial frames of reference, which are not accelerating.

• Example: A hockey puck gliding on frictionless ice remains in motion unless a force acts on it.

Newton's Second Law of Motion

Force and Acceleration

Newton's second law quantifies the relationship between force, mass, and acceleration. The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Mathematical Statement: $\sum \vec{F} = m \vec{a}$

• Direct Proportionality: Increasing the net force increases acceleration.

• Inverse Proportionality: Increasing the mass decreases acceleration for the same net force.

• Unit: $1\,\text{N} = 1\,\text{kg} \cdot \text{m/s}^2$

• Example: Doubling the net force on an object doubles its acceleration; doubling the mass halves its acceleration for the same force.

Mass and Weight

Distinction Between Mass and Weight

Mass is a measure of the amount of matter in an object and is independent of location. Weight is the gravitational force acting on the object and depends on the local acceleration due to gravity.

• Mass: Scalar quantity, measured in kilograms (kg).

• Weight: Vector quantity, measured in newtons (N).

• Formula: $w = m g$

• Example: An object with mass $2\,\text{kg}$ has a weight of $19.6\,\text{N}$ on Earth ($g = 9.8\,\text{m/s}^2$).

• Additional info: The value of $g$ varies with altitude and location; on other planets, $g$ is different.

Newton's Third Law of Motion

Action and Reaction

Newton's third law states that for every action, there is an equal and opposite reaction. When two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction.

• Mathematical Statement: $\vec{F}_{AB} = -\vec{F}_{BA}$

• Interaction: Forces always occur in pairs.

• Example: When you push against a wall, the wall pushes back with an equal and opposite force.

• Application: Walking depends on pushing backward on the ground; the ground pushes forward, propelling you ahead.

Free-Body Diagrams

Visualizing Forces

A free-body diagram is a graphical representation used to visualize all the forces acting on a single object. It is a crucial tool for solving problems in dynamics.

• Purpose: To identify and analyze all forces acting on an object.

• Components: The object is represented as a dot or simple shape; arrows show all forces acting on it.

• Example: A block on a surface may have arrows for gravity (downward), normal force (upward), friction (horizontal), and any applied force.