Newton’s Laws of Motion – Study Notes

Newton’s Laws of Motion

Introduction to Dynamics

Dynamics is the branch of physics concerned with 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 one, two, or three dimensions.

• Dynamics relates motion to its causes—forces.

• Newton’s laws were deduced from experimental observations.

Properties of a Force

In physics, a force is a fundamental concept that describes an interaction capable of changing an object’s motion. Forces are always vectors, meaning they have both magnitude and direction.

• 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 Nature: Forces are vector quantities, represented by arrows indicating direction and length proportional to magnitude.

• Notation: The symbol $\vec{F}$ is commonly used to denote force vectors.

• Example: Pushing a box (force vector points away from the hand), pulling a box (force vector points toward the hand).

Forces and the Particle Model

To analyze forces acting on objects, physicists often use the particle model, treating the object as a point mass and representing forces as vectors acting at that point.

• Model the object as a particle.

• Place the tail of the force vector on the particle.

• Draw the force vector as an arrow pointing in the direction of the force, with length proportional to its magnitude.

• Label the vector appropriately (e.g., $\vec{F}$).

• Example: Drawing force vectors for a box being pushed, pulled, or acted on by gravity.

Vector Representation of Forces

Forces can be represented as vectors in diagrams to visualize their effects on objects. Translated forces with equal length and direction are considered equal, regardless of their position.

• Always draw the tail of the force vector at the particle.

• Equal forces have equal magnitude and direction, even if drawn at different locations.

• Example: Pushing force of rope, pushing force of spring, long-range force of gravity.

Common Types of Forces

Several types of forces commonly appear in physics problems. Each has distinct characteristics and arises from different physical interactions.

• Normal Force ($\vec{N}$): The perpendicular contact force exerted by a surface on an object resting on it.

• Friction Force ($\vec{f}$): The parallel contact force exerted by a surface, opposing relative motion.

• Tension Force ($\vec{T}$): The pulling force exerted by a rope, cable, or string.

• Gravity ($\vec{F}_g$): The long-range attractive force between objects with mass, most commonly the Earth’s pull on objects.

• Elastic Force: The restoring force exerted by a spring or elastic material, described by Hooke’s Law: $F = -k x$.

• Drag Force: The resistive force exerted by a fluid (air or water) on a moving object.

• Buoyant Force: The upward force exerted by a fluid, opposing the weight of an immersed object.

• Thrust: The reaction force produced when a system expels or accelerates mass (e.g., rocket propulsion).

Units of Force

The SI unit of force is the newton (N). Other systems use pounds (British) or dynes (cgs).

• 1 newton (N): The force required to accelerate a 1 kg mass by 1 m/s2.

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

Superposition of Forces

When multiple forces act on an object, their effects combine according to the principle of superposition. The net force is the vector sum of all individual forces.

• Resultant Force: $\vec{R} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \ldots$

• The net force determines the object’s acceleration.

Decomposing Forces into Components

Forces can be broken down into perpendicular components, typically along the x- and y-axes, using trigonometry.

• Component Formulas:

  • $F_x = F \cos \theta$

  • $F_y = F \sin \theta$

• Useful for analyzing forces in two dimensions.

Newton’s First Law of Motion (Law of Inertia)

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. This law defines the concept of equilibrium.

• Equilibrium Condition: $\sum \vec{F} = 0$

• If the net force is zero, acceleration is zero.

• Valid only in inertial frames of reference (non-accelerating frames).

• Example: A sled sliding at constant velocity on ice, with all forces balanced.

Inertial and Non-Inertial Frames of Reference

Newton’s laws are valid only in inertial frames—frames that are either at rest or moving at constant velocity. Accelerating frames are non-inertial and require fictitious forces for analysis.

• Inertial Frame: No acceleration; Newton’s laws apply directly.

• Non-Inertial Frame: Accelerating; apparent forces (e.g., feeling pushed back in a bus) are observed.

• Example: Crash test dummies appear to be pushed forward in a stopping car due to inertia.

Newton’s Second Law of Motion

The acceleration of an object is directly proportional to the net external force acting on it and inversely proportional to its mass.

• Formula: $\sum \vec{F} = m \vec{a}$

• Increasing net force increases acceleration; increasing mass decreases acceleration.

• Example: Applying a constant force to two objects of different mass results in different accelerations.

Mass and Weight

Mass is a measure of the amount of matter in an object and its resistance to acceleration (inertial mass). Weight is the gravitational force exerted on an object by the Earth.

• Weight Formula: $w = m g$

• g: Acceleration due to gravity (approximately $9.8\,\text{m/s}^2$ on Earth).

• Mass is constant; weight depends on location (e.g., different planets).

• Example: A 10 kg object has a weight of $98\,\text{N}$ on Earth.

Newton’s Third Law of Motion

For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.

• Formula: $\vec{F}_{A \to B} = -\vec{F}_{B \to A}$

• Action-reaction pairs are always the same type of force and act on different objects.

• Example: When you kick a ball, the ball exerts an equal force back on your foot.

Defining the System

When analyzing forces, it is important to define the system of interest. Internal forces (between parts of the system) cancel out, while external forces affect the system’s motion.

• Draw a boundary around the system to distinguish internal and external forces.

• Only external forces contribute to the net force and acceleration of the system.

• Example: In a cart-rope-apple system, tension forces are internal and cancel out.

Free-Body Diagrams

A free-body diagram is a graphical representation showing all the forces acting on a single object. It is a crucial tool for solving problems involving Newton’s laws.

• Represent the object as a dot or simple shape.

• Draw all external forces acting on the object as arrows.

• Label each force clearly (e.g., gravity, normal, friction, tension).

• Use the diagram to set up equations for equilibrium or acceleration.

• Example: Free-body diagram for a block on a table includes gravity downward and normal force upward.

Summary Table: Common Types of Forces

Force Type	Direction	Contact/Long-Range	Example
Normal ($\vec{N}$)	Perpendicular to surface	Contact	Book resting on table
Friction ($\vec{f}$)	Parallel to surface, opposes motion	Contact	Sliding box
Tension ($\vec{T}$)	Along rope, away from object	Contact	Hanging mass
Gravity ($\vec{F}_g$)	Toward center of Earth	Long-range	Falling apple
Elastic	Restores to equilibrium	Contact	Compressed spring
Drag	Opposes motion through fluid	Contact	Skydiver in air
Buoyant	Upward in fluid	Contact	Floating object
Thrust	In direction opposite expelled mass	Contact/Long-range	Rocket propulsion

Additional info: Some explanations and examples have been expanded for clarity and completeness, based on standard physics curriculum.