Applications of Newton's Laws of Motion

Applications of Newton's Laws of Motion

The application of Newton's Laws to analyze forces and predict motion in mechanical systems.

Free-Body Diagram

A single body or subsystem isolated from its surroundings showing all external forces acting on it.

Free-Body Diagram Purpose

Allows convenient application of Newton's Second Law to mechanical systems.

External Forces

Forces acting on an object from its surroundings.

Object of Interest

The body or subsystem selected for analysis in a free-body diagram.

Boundary

The clearly defined limit separating the object or system from its surroundings.

Drawing a Free-Body Diagram Step 1

Identify the object or system and isolate it from other objects while clearly specifying its boundary.

Drawing a Free-Body Diagram Step 2

Draw the non-contact external force first, which is generally the weight.

Drawing a Free-Body Diagram Step 3

Draw the contact forces acting at the boundary of the object or system.

Non-Contact Force

A force that acts without physical contact.

Weight

The non-contact force usually drawn first in a free-body diagram.

Contact Forces

Normal force, friction, tension, and applied force.

Normal Force

A contact force exerted perpendicular to the surface.

Friction Force

A contact force that opposes relative motion between surfaces.

Tension

A pulling force transmitted through a rope, string, or cable.

Applied Force

A force exerted directly on an object by another object or person.

Newton's Second Law

Relates the net force acting on an object to its mass and acceleration.

Newton's Second Law Formula

ΣF = ma

Net Force

The vector sum of all forces acting on an object.

Net Force Symbol

ΣF

Force Analysis

The process of resolving forces into their x- and y-components before applying Newton's Second Law.

Horizontal Force Equation

ΣFx = max

Vertical Force Equation

ΣFy = may

Horizontal Equilibrium

ΣFx = 0

Vertical Equilibrium

ΣFy = 0

Equilibrium Condition

The net force acting on the object is zero.

Acceleration

The result of a nonzero net force acting on an object.

Weight Symbol

W

Weight Formula

W = mg

Normal Force Symbol

N

Tension Symbol

T

Acceleration Symbol

a

Mass Symbol

m

Gravitational Acceleration Symbol

g

Coordinate Axes

The x- and y-axes chosen for force analysis.

Atwood Machine

A system of masses connected by a rope passing over a pulley.

Free-Body Diagram for an Atwood Machine

Shows only the external forces acting on each mass separately.

Weight on Mass 1

W₁

Weight on Mass 2

W₂

Tension in an Ideal Rope

The same throughout a massless, frictionless rope.

Equation of Motion

A mathematical equation relating the forces acting on a system to its acceleration.

Mechanical System

A collection of bodies whose motion is analyzed using Newton's Laws.

Force Components

The horizontal and vertical parts of a force vector.

Analysis of a Free-Body Diagram

The application of ΣFx = max and ΣFy = may to determine unknown quantities.

Inclined Plane

A flat surface tilted at an angle to the horizontal.

Angle of Inclination

The angle between the inclined plane and the horizontal.

Angle of Inclination Symbol

θ

Weight on an Inclined Plane

The gravitational force acting vertically downward.

Weight Components

The components of weight parallel and perpendicular to the inclined plane.

Parallel Component of Weight

W∥ = mg sin θ

Perpendicular Component of Weight

W⊥ = mg cos θ

Normal Force on an Inclined Plane

N = mg cos θ (when no other vertical forces act).

Force Parallel to the Incline

The component of force responsible for accelerating the object along the slope.

Force Perpendicular to the Incline

The component balanced by the normal force.

Coordinate System on an Inclined Plane

The x-axis is chosen parallel to the incline and the y-axis perpendicular to the incline.

Newton's Second Law Along the Incline

ΣFx = max

Newton's Second Law Perpendicular to the Incline

ΣFy = 0

Object Sliding Down an Incline

Acceleration is caused by the component of gravity parallel to the incline.

Object at Rest on an Incline

The net force parallel to the incline is zero.

Friction on an Inclined Plane

Acts parallel to the surface and opposes the direction of motion or impending motion.

Static Friction on an Incline

Prevents motion until the maximum static friction is exceeded.

Kinetic Friction on an Incline

Acts while the object is sliding.

Static Friction Formula

fs ≤ μsN

Maximum Static Friction Formula

fs(max) = μsN

Kinetic Friction Formula

fk = μkN

Coefficient of Static Friction Symbol

μs

Coefficient of Kinetic Friction Symbol

μk

Connected Bodies

Two or more objects joined by a rope, string, or cable.

Ideal Rope

A rope that is massless and inextensible.

Ideal Pulley

A pulley that is massless and frictionless.

Tension in an Ideal Rope

The tension is the same throughout the rope.

Acceleration of Connected Bodies

All connected bodies have the same magnitude of acceleration.

Direction of Tension

Tension always pulls away from the object.

System Analysis

Treating connected objects as a single system to simplify calculations.

Internal Forces

Forces between objects within a system that cancel when the entire system is analyzed.

External Forces

Forces acting on the system from outside.

Net External Force

The total external force acting on the system.

Free-Body Diagram for Connected Bodies

Draw a separate free-body diagram for each object.

Common Acceleration

All connected objects move with the same acceleration.

Force Balance

Apply Newton's Second Law separately to each object.

Positive Direction

Choose one direction consistently before writing equations.

Acceleration on an Inclined Plane

a = ΣF/m

Weight Component Causing Motion

mg sin θ

Weight Component Balanced by the Normal Force

mg cos θ

Limiting Static Friction

The maximum friction before an object starts to move.

Motion on an Incline

Begins when the parallel component of weight exceeds the maximum static friction.

Analysis of Free-Body Diagrams

The process of applying Newton's Second Law to a free-body diagram to determine unknown quantities.

Purpose of Free-Body Diagram Analysis

To apply ΣF = ma in mechanical systems.

Newton's Second Law in Mechanical Systems

ΣF = ma

Horizontal Force Equation

ΣFx = max

Vertical Force Equation

ΣFy = may

Horizontal Analysis

Apply Newton's Second Law along the x-axis.

Vertical Analysis

Apply Newton's Second Law along the y-axis.

Acceleration Along an Axis

Determined by the net force acting along that axis.

Equation of Motion

An equation relating the acceleration of a system to the forces acting on it.

Atwood's Machine

A system consisting of masses connected by an ideal rope over a pulley.

Ideal Pulley

A pulley assumed to be massless and frictionless.

Ideal Rope

A rope assumed to be massless and inextensible.

Common Acceleration

All connected objects move with the same magnitude of acceleration.

Equal Tension

The tension is the same throughout an ideal rope.