Physics 111: Chapter 4 – Laws of Motion (Study Notes)

Forces

Definition and Nature of Forces

Forces are fundamental interactions that cause changes in the motion of objects. In physics, a force is defined as any interaction that, when unopposed, will change the motion of an object. Forces are vector quantities, meaning they have both magnitude and direction.

• Contact forces arise from physical contact between objects (e.g., friction, tension).

• Field forces act over a distance without direct contact (e.g., gravitational, electromagnetic).

Examples of forces:

• Gravitational Force

• Friction Force

• Tension Force

• Normal Force

Units and Vector Components

• The SI unit of force is the Newton (N):

$1\,N = 1\,\dfrac{kg \cdot m}{s^2}$

• Forces can be measured by the deformation of a spring (Hooke's Law).

• Forces are vectors and can be decomposed into x and y components:

$\vec{F} = F_x \hat{i} + F_y \hat{j}$

• The x and y components of a force are independent of each other.

• Motion in the x-direction is affected only by the x-component of the force, and similarly for the y-direction.

Newton's Laws of Motion

Newton's First Law (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.

$\vec{F} = 0 \implies \vec{v} = \text{constant}$

• This law introduces the concept of inertia, the tendency of objects to resist changes in their state of motion.

• Example: A block moving at constant velocity on a frictionless surface will continue moving unless a force acts on it.

Conceptual Questions

• An object can be in motion in the absence of forces (true in the absence of friction or other forces).

• An object at rest can be subjected to forces (e.g., balanced forces resulting in no motion).

Example Problem

• If a block moves with constant velocity across a horizontal surface, the forces acting on it (F1 and F2) must be equal in magnitude and opposite in direction, resulting in zero net force and zero acceleration.

Newton's Second Law

Newton's Second Law quantifies the relationship between force, mass, and acceleration. It states that the net force acting on an object is equal to the product of its mass and acceleration.

$\vec{F}_{\text{net}} = m \vec{a}$

• Mass is a measure of an object's resistance to changes in velocity (inertia).

• The SI unit of mass is the kilogram (kg).

• Forces in different directions are treated independently:

$F_x = m a_x$ $F_y = m a_y$

• When multiple forces act on an object, the net force is the vector sum of all individual forces.

Free Body Diagrams

• A free body diagram is a crucial tool for analyzing forces acting on an object. It shows all external forces acting on the object of interest.

• Only include forces acting on the object, not forces the object exerts on others.

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. If object 1 exerts a force on object 2, then object 2 exerts a force of equal magnitude and opposite direction on object 1.

$\vec{F}_{12} = -\vec{F}_{21}$

• These forces act on different objects and do not cancel each other out.

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

Mass and Inertia

Definition of Mass

Mass is a property of matter that quantifies the amount of inertia an object possesses. It determines how much an object resists acceleration when a force is applied.

• Objects with greater mass are harder to accelerate.

• The SI unit of mass is the kilogram (kg).

Types of Forces

Gravitational Force

The gravitational force is the attractive force between any two masses. Near Earth's surface, it gives objects weight.

$F_g = m g$

• Where $g \approx 9.8\,m/s^2$ on Earth.

• Direction: vertically downward.

• The SI unit of weight is the Newton (N).

Normal Force

The normal force is the support force exerted by a surface perpendicular to the object resting on it. It prevents objects from falling through surfaces.

• Direction: always perpendicular to the surface.

• Magnitude may not always equal $mg$ (depends on other forces and orientation).

Tension Force

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends.

• Direction: always along the rope and away from the object.

• If the rope is massless, the tension is the same throughout its length.

Equilibrium and Non-Equilibrium

Equilibrium

An object is in equilibrium if it is at rest or moving with constant velocity. In this state, the net force acting on the object is zero.

$\sum \vec{F} = 0$

• All forces balance each other.

• Acceleration is zero.

Non-Equilibrium

If the net force on an object is not zero, the object will accelerate in the direction of the net force.

$\sum \vec{F} = m \vec{a}$

• Examples include objects speeding up, slowing down, or changing direction.

Summary Table: Types of Forces

Type of Force	Description	Direction	Formula (if applicable)
Gravitational	Attraction between masses	Downward (toward center of Earth)	$F_g = m g$
Normal	Support force from a surface	Perpendicular to surface	Varies (often $N = mg$ on flat surface)
Tension	Force in a stretched string/rope	Along the string, away from object	Depends on system
Friction	Opposes motion between surfaces	Parallel to surface, opposite to motion	$f = \mu N$ (static/kinetic friction)

Key Concepts and Applications

• Always draw a free body diagram to analyze forces.

• Use Newton's laws to relate forces and motion.

• Remember that forces in perpendicular directions are independent.

• In equilibrium, net force is zero; in non-equilibrium, net force causes acceleration.