# Statics

## Overview

Welcome to the in-depth course material on Statics in Further Mathematics, focusing on Vectors and Mechanics. In this study, we delve into the fundamental concepts of statics, which form the basis for understanding the equilibrium of forces and applying vector algebra in solving static problems.

Statics deals with the analysis of objects at rest or moving at constant velocity under the action of various forces. To grasp statics effectively, it is crucial to differentiate between scalar and vector quantities. Scalars are quantities that have only magnitude, such as mass, while vectors possess both magnitude and direction, like force.

Understanding the algebra of vectors is pivotal in statics. Vectors exhibit properties like commutativity, associativity, and distributivity, which are essential for manipulating vector quantities. Furthermore, unit vectors help in expressing any vector in terms of its components along the coordinate axes, aiding in vector operations.

Representation of vectors is key to visualizing forces acting on bodies. Position vectors indicate the location of a point relative to a reference point or origin. These vectors assist in determining distances and directions in statics problems, contributing to the overall analysis of forces.

Resolution and composition of vectors are fundamental skills in statics, enabling the breakdown of vectors into perpendicular components or the combination of vectors to find their resultant. This process aids in simplifying complex force systems and determining the net effect of multiple forces acting simultaneously.

Scalar product, also known as the dot product, involves multiplying the magnitudes of two vectors by the cosine of the angle between them, resulting in a scalar quantity. This product finds applications in calculating work done by a force or determining the projection of one vector onto another in statics.

On the other hand, vector product, or cross product, produces a vector perpendicular to the plane containing two input vectors. This operation is crucial for determining moments of forces in static equilibrium situations, providing insights into the rotational effects of forces on rigid bodies.

Moreover, the course material covers the definition of a force, its representation through vectors, and the significance of coplanar forces acting at a point. By applying principles of equilibrium, students learn how to balance forces and torques to maintain the stability of bodies in static situations.

Lastly, the concepts of friction play a vital role in statics, particularly in distinguishing between smooth and rough planes. Determining the coefficient of friction allows for the analysis of forces resisting motion and helps in predicting the behavior of objects on various surfaces.

## Objectives

1. Apply lami's theorem and principles of moments to solve problems in statics
2. Apply the principles of statics to solve problems involving forces and equilibrium
3. Apply vector algebra in solving problems involving static equilibrium
4. Differentiate between scalar and vector quantities in statics
5. Determine the coefficient of friction in statics problems
6. Determine the resultant of forces acting on rigid bodies in statics problems
7. Understand the distinction between smooth and rough planes in statics
8. Understand the concepts of statics in Further Mathematics

## Lesson Note

Statics is a branch of mechanics that deals with bodies at rest or in equilibrium. Understanding statics is essential in various fields such as engineering, physics, and architecture, as it allows us to analyze and predict the forces acting on a structure or object.

## Lesson Evaluation

Congratulations on completing the lesson on Statics. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.

You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.

Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.

1. Define a force in mechanics. A. A scalar quantity B. A quantity that causes an object to move C. A vector quantity D. The resistance of an object to change in motion Answer: C. A vector quantity
2. Which property of vectors states that a + b = b + a? A. Distributive property B. Commutative property C. Associative property D. Vector property Answer: B. Commutative property
3. What is the unit vector for a vector v = 3i + 4j? A. (3, 4) B. (1, 1) C. (3i, 4j) D. (3/5, 4/5) Answer: D. (3/5, 4/5)
4. If two forces of magnitudes 5N and 12N are acting at an angle of 60 degrees to each other. What is the resultant force magnitude? A. 15N B. 17N C. 20N D. 13N Answer: B. 17N
5. What type of equilibrium occurs when an object is at rest and has no rotational motion? A. Static equilibrium B. Dynamic equilibrium C. Unstable equilibrium D. Neutral equilibrium Answer: A. Static equilibrium

## Past Questions

Wondering what past questions for this topic looks like? Here are a number of questions about Statics from previous years

Question 1

A body of mass of 18kg is suspended by an inextensible string from a rigid support and is pulled by a horizontal force F until the angle of inclination of the string to the vertical is 35º. If the system is in equilibrium, calculate the:

i. value of F

ii. tension in the string

Practice a number of Statics past questions