Chapter 6: Force and Motion II

Overview

This chapter builds upon the foundation of forces and motion from Form 4 by introducing more advanced concepts including resultant forces, force resolution, conditions of equilibrium, and elasticity. These concepts are essential for understanding complex mechanical systems, structural engineering, and various applications in physics and engineering.

Learning Objectives

After completing this chapter, you will be able to:

Calculate resultant forces using vector addition
Resolve forces into perpendicular components
Apply conditions for equilibrium in static and dynamic situations
Understand and apply Hooke's Law
Calculate elastic potential energy
Analyze force systems in equilibrium

Resultant Force

Main Concept

A resultant force is a single force that represents the combined effect of two or more forces acting on an object. It is determined through vector addition.

Key Principles

Triangle of Forces: If three forces acting at a point are in equilibrium, they can be represented by the sides of a triangle drawn in order.
Parallelogram of Forces: If two forces are represented by two adjacent sides of a parallelogram, the diagonal through the point of intersection represents the resultant force.

Key Formulas

For two perpendicular forces $F_x$ and $F_y$ :

Magnitude of Resultant Force, $F = \sqrt{F_x^2 + F_y^2}$
Direction, $θ = \tan^{-1}(F_y / F_x)$

Important Terms

Vector Addition: Process of combining vector quantities

Did You Know?

The concept of resultant forces is crucial in structural engineering. Bridges, buildings, and other structures must be designed so that the resultant force on any part is zero for stability.

Resultant Force Diagrams

Force Vector Addition Visualization

Force Resolution

Main Concept

A single force can be resolved into two perpendicular components (usually horizontal and vertical components).

Key Principles

Resolving forces allows analysis of motion on inclined planes and situations where forces act at an angle.
This technique is essential for breaking down complex force systems into manageable components.

Key Formulas

If force $F$ acts at angle $θ$ with the horizontal:

Horizontal Component, $F_x$ : $F_x = F \cos(θ)$
Vertical Component, $F_y$ : $F_y = F \sin(θ)$

Important Terms

Component: Part of a vector in a specific direction
Inclined Plane: Flat surface tilted at an angle

Force Resolution Diagrams

Component Analysis Visualization

Worked Example

Problem: A force of 100 N acts at an angle of 30° above the horizontal. Resolve this force into horizontal and vertical components.

Solution:

Force, F = 100 N
Angle, θ = 30°

Horizontal component:

F_x = F \cos(θ) = 100 \cos(30°) = 100 \times 0.866 = 86.6 \text{ N}

Vertical component:

F_y = F \sin(θ) = 100 \sin(30°) = 100 \times 0.5 = 50 \text{ N}

Answer: Horizontal component = 86.6 N, Vertical component = 50 N

Force Resolution Example Diagram

Forces in Equilibrium

Main Concept

An object is said to be in force equilibrium when the resultant force acting on it is zero. The object is either at rest (static equilibrium) or moving with constant velocity (dynamic equilibrium).

Key Principles

When in equilibrium, $\Sigma F = 0$ . This means the sum of all upward forces equals the sum of all downward forces, and the sum of all leftward forces equals the sum of all rightward forces.

Key Formulas

$\Sigma F_x = 0$
$\Sigma F_y = 0$

Important Terms

Static Equilibrium: Object is at rest
Dynamic Equilibrium: Object is moving with constant velocity

Equilibrium Conditions

For an object to be in equilibrium:

The resultant force must be zero ( $\Sigma F = 0$ )
The resultant torque must be zero (for rotational equilibrium)

Practical Applications

Bridge Design: Engineers ensure all forces are balanced
Building Construction: Structures must maintain equilibrium under load
Cranes: Use equilibrium principles to lift heavy loads safely

Equilibrium System Diagrams

Force Vector Equilibrium

Elasticity

Main Concept

Elasticity is the property of materials that allows them to return to their original shape and size after the force acting on them is removed.

Key Principles

Hooke's Law: The extension of a spring is directly proportional to the force applied, provided the elastic limit is not exceeded.

Key Formulas

Hooke's Law, F:

F = kx

Elastic Potential Energy, $E_p$ :

E_p = \frac{1}{2}Fx \text{ or } E_p = \frac{1}{2}kx^2

Where:

$F$ = Force (N)
$k$ = Spring constant (N m⁻¹)
$x$ = Extension or compression (m)
$E_p$ = Elastic potential energy (J)

Important Terms

Elastic Limit: Maximum force that can be applied to a material before it undergoes permanent deformation
Spring Constant, k: Measure of spring stiffness. Unit: N m⁻¹

Elastic Properties Table

Material	Spring Constant (N m⁻¹)	Elastic Limit (N)
Steel	High (2000-20000)	Very High
Copper	Medium (100-1000)	Medium
Rubber	Low (1-100)	Low
Plastic	Variable	Low

Elasticity Diagrams

Hooke's Law Visualization

Hooke's Law Experiments

Key Concepts

Linear Region: Where Hooke's Law applies (force proportional to extension)
Elastic Limit: Beyond which permanent deformation occurs
Plastic Region: Where materials deform permanently

Experimental Setup

Apparatus: Spring, masses, ruler, retort stand
Procedure: Add masses gradually and measure extension
Graph: Plot force vs extension to determine spring constant

Graph Analysis

Slope: Represents spring constant (k)
Elastic Limit: Point where graph curves
Hooke's Law Valid: Straight line through origin

Applications of Elasticity

Real-World Examples

Suspension Systems: Car springs absorb shocks
Bungee Jumping: Elastic cords provide safety
Mouse Traps: Use elastic energy
Sports Equipment: Tennis rackets, bows
Engineering: Vibration dampers, shock absorbers

Safety Considerations

Material Selection: Choose appropriate materials for load conditions
Overloading: Avoid exceeding elastic limits
Fatigue: Materials weaken under repeated stress

SPM Exam Tips

Common Mistakes to Avoid

Vector Confusion: Remember forces are vectors with magnitude and direction
Component Resolution: Always resolve forces properly before adding
Equilibrium Conditions: Both ΣF_x = 0 and ΣF_y = 0 must be satisfied
Units: Always use correct units (N for force, m for extension)

Problem-Solving Strategies

Draw Free Body Diagram: Sketch all forces acting on the object
Choose Coordinate System: Usually horizontal and vertical
Resolve Forces: Break down diagonal forces into components
Apply Equilibrium Conditions: Set ΣF_x = 0 and ΣF_y = 0
Solve Equations: Find unknown forces or angles

Important Formula Summary

Concept	Formula
Resultant Force	F = √(F_ $x^2$ + F_ $y^2$ )
Force Components	F_x = F cos(θ), F_y = F sin(θ)
Equilibrium	ΣF_x = 0, ΣF_y = 0
Hooke's Law	F = kx
Elastic Energy	E_p = ½k $x^2$

Practice Questions

A 5 kg object is suspended by two strings making angles of 30° and 60° with the vertical. Calculate the tension in each string.
A force of 200 N acts at an angle of 45° to the horizontal. Resolve this force into horizontal and vertical components.
A spring extends by 2 cm when a 10 N weight is hung from it. Calculate the spring constant and the elastic potential stored.
A ladder leaning against a wall is in equilibrium. If the ladder weighs 200 N and makes an angle of 60° with the ground, calculate the normal forces at the wall and ground.
Explain why a spring balance shows different readings when moved up and down in an accelerating elevator.

Summary

This chapter covered essential concepts in advanced force and motion:

Resultant Forces: Combining multiple forces into a single equivalent force
Force Resolution: Breaking forces into perpendicular components
Equilibrium: Conditions for balanced force systems
Elasticity: Hooke's Law and elastic potential energy
Applications: Real-world uses of these principles

Master these concepts to understand complex mechanical systems, structural engineering, and material properties - fundamental to many physics and engineering applications.