Unit 1: Introduction to Mechanics

1. Basic Concepts

Engineering Mechanics is the branch of physical science that deals with the behavior of bodies at rest or in motion when subjected to forces. It forms the foundation of all structural and mechanical design.

Divisions of Mechanics

• Statics: The study of bodies in equilibrium (at rest or moving with a constant velocity) under the action of forces.
• Dynamics: The study of bodies in motion under the action of forces. It is further divided into:
  • Kinematics: Study of motion without considering the forces causing it (focus on displacement, velocity, acceleration).
  • Kinetics: Study of motion and the forces that cause it.

Fundamental Terms

• Space: The geometric region in which events occur, described by linear and angular measurements relative to a coordinate system.
• Time: The measure of the succession of events.
• Mass: The quantity of matter in a body. It provides a measure of the body's inertia (resistance to change in motion).
• Force: An action of one body on another that tends to move or deform it. It is a vector quantity characterized by magnitude, direction, and point of application.
• Particle: A body with mass but whose dimensions are negligible.
• Rigid Body: A body in which the relative distance between any two internal points remains constant, regardless of applied forces (no deformation occurs).

2. System of Forces

When two or more forces act on a body, they constitute a force system. Force systems are classified based on the lines of action of the forces:

• Coplanar Forces: The lines of action of all forces lie in the same two-dimensional plane.
• Non-Coplanar (Space) Forces: The lines of action of forces lie in three-dimensional space.
• Concurrent Forces: The lines of action of all forces intersect at a single common point.
• Non-Concurrent Forces: The lines of action do not intersect at a single point.
• Collinear Forces: The lines of action of all forces lie along the same straight line.
• Parallel Forces: The lines of action of all forces are parallel to each other. Can be like (same direction) or unlike (opposite directions).

3. Coplanar Concurrent Forces

Coplanar concurrent forces lie in the same plane and meet at a single point. The fundamental laws governing these forces include:

Parallelogram Law of Forces

If two forces acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal passing through their point of intersection.

• Magnitude of Resultant ():
• Direction of Resultant ():
  (where is the angle between forces and , and is the angle makes with )

Triangle Law of Forces

If two forces acting on a particle are represented by the two sides of a triangle taken in order, their resultant is represented by the third side of the triangle taken in the opposite order.

Polygon Law of Forces

If a number of concurrent forces are represented by the sides of a polygon taken in order, their resultant is represented by the closing side of the polygon taken in the opposite order.

4. Components in 2-D Plane & Resultant

Resolution of Forces

Resolution is the process of splitting a single force into two or more components without changing its effect on the body. Usually, forces are resolved into mutually perpendicular (rectangular) components along the x and y axes.

• If a force makes an angle with the x-axis:
  • Horizontal component:
  • Vertical component:

Resultant of a Coplanar Concurrent System

To find the resultant of multiple concurrent forces:

1. Resolve all forces into their x and y components.
2. Sum the horizontal components algebraically:
3. Sum the vertical components algebraically:
4. Calculate the magnitude of the Resultant ():
   
5. Calculate the direction () of the resultant relative to the x-axis:
   

5. Moment of Forces and its Applications

Moment of a Force: The measure of the rotational tendency (or turning effect) of a force about a specific point or axis.

• Formula:
  (where is the magnitude of the force, and is the perpendicular distance from the axis of rotation to the line of action of the force)
• Units: Newton-meters (N·m).
• Sign Convention: Counter-clockwise (CCW) is typically positive (+), and clockwise (CW) is negative (-).

Varignon’s Theorem (Principle of Moments)

The algebraic sum of the moments of a system of coplanar concurrent forces about any point in their plane is equal to the moment of their resultant force about the same point.

• Formula:

Applications:

• Determining the point of application of the resultant of parallel or non-concurrent forces.
• Calculating reactions at supports in beams.
• Designing levers and mechanical linkages.

6. Couples and Resultant of Force System

Couples

A couple consists of two forces that are equal in magnitude, opposite in direction, and whose lines of action are parallel but do not coincide.

• Characteristics of a Couple:
  • The translational resultant is zero ().
  • It produces purely rotational motion.
  • The moment of a couple is constant about any point in the plane.
• Moment of a Couple ():
  (where is the magnitude of one of the forces, and is the perpendicular distance between their lines of action)

Resolution of a Force into a Force and a Couple

Any force acting at point A can be moved to a parallel position at point B by adding a couple , where is the perpendicular distance between A and B.

Resultant of a Coplanar Non-Concurrent Force System

To find the resultant of a general coplanar system:

1. Find and .
2. Calculate resultant magnitude .
3. Calculate the sum of moments about a chosen point : .
4. Find the perpendicular distance from point to the line of action of : .

7. Free Body Diagrams (FBD)

A Free Body Diagram (FBD) is a simplified sketch of a body (or part of a body) isolated from its surroundings, showing all the external forces and reactive forces acting upon it.

Importance of FBD

• It visually identifies all forces acting on a system.
• It forms the basis for writing the equations of equilibrium accurately.

Steps to Draw an FBD

1. Isolate the Body: Draw the outline of the body detached from all external supports and other bodies.
2. Add Applied Forces: Indicate all known external loads, weights, and applied moments. (Weight always acts vertically downward through the center of gravity).
3. Add Reactions: Replace supports with their corresponding reactive forces/moments:
   • Smooth Surface/Roller: One perpendicular reaction force.
   • Hinge/Pin: Two reaction forces (horizontal and vertical).
   • Fixed Support: Two reaction forces (horizontal and vertical) and one resisting moment.
   • Cable/String: Tension force acting away from the body along the cable.
4. Label: Assign variable names to unknown forces/dimensions and write down known values.

8. Equilibrium of System of Forces & Equations of Equilibrium

Concept of Equilibrium

A body is said to be in equilibrium if it remains at rest or moves with a constant velocity. In equilibrium, the resultant of all forces and the resultant of all moments acting on the body are zero.

Lami's Theorem

If three coplanar, concurrent forces acting on a particle keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

• Formula:
  (where are the forces and are the angles opposite to them respectively)

Equations of Equilibrium for Co-planar Systems

Depending on the type of force system, different sets of equilibrium equations apply:

1. Coplanar Collinear System:
   • (assuming forces are along the x-axis)
2. Coplanar Concurrent System:
   • 
     (No moment equation is needed because all forces pass through the same point, generating zero moment).
3. Coplanar Parallel System:
   • (Sum of forces parallel to the common axis)
   • (Sum of moments about any point in the plane)
4. Coplanar Non-Concurrent (General) System:
   To achieve complete equilibrium, a body must have neither translational nor rotational acceleration.
   • (about any point)