1.1 System of Coplanar Forces:

### Classification of Force Systems:

**Concurrent Forces:**

- Resultant Force: \( \mathbf{R} = \sum \mathbf{F} \)
- Moment About a Point: \( \mathbf{M} = \sum \mathbf{r} \times \mathbf{F} \)

**Parallel Forces:**

- Resultant Force: \( R = \sum F_i \)
- Location of Resultant: \( x = \frac{\sum F_i \cdot x_i}{\sum F_i} \)

**Non-concurrent Non-parallel Forces:**

- Resultant Force: \( \mathbf{R} = \sqrt{\left(\sum F_x\right)^2 + \left(\sum F_y\right)^2} \)
- Angle of Resultant: \( \tan \theta = \frac{\sum F_y}{\sum F_x} \)

### Principle of Transmissibility:

The effect of a force on a rigid body is the same whether the force is applied at its original point or along its line of action.

### Composition and Resolution of Forces:

\( \mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2 + \ldots + \mathbf{F}_n \)

\( \mathbf{F}_x = \sum \mathbf{F}_i \cos \theta_i \)

\( \mathbf{F}_y = \sum \mathbf{F}_i \sin \theta_i \)

## 1.2 Resultant:

### Resultant of Coplanar and Non-Coplanar Force Systems:

**Concurrent Forces:** \( \mathbf{R} = \sum \mathbf{F}_i \)

**Parallel Forces:** \( R = \sum F_i \)

**Non-concurrent Non-parallel Forces:** \( \mathbf{R} = \sqrt{\left(\sum F_x\right)^2 + \left(\sum F_y\right)^2} \)

### Moment of Force about a Point:

\( \mathbf{M} = \mathbf{r} \times \mathbf{F} \)

### Couples and Varignon's Theorem:

Couples: \( \mathbf{M} = \mathbf{F} \cdot d \)

Varignon's Theorem: \( \mathbf{R} = \sqrt{\mathbf{F}_1^2 + \mathbf{F}_2^2} \)

### Distributed Forces in Plane:

For distributed loads, integrate to find resultant forces and moments.

## Centroid:

### First Moment of Area:

\( Q_x = \int_A x \, dA \)

\( Q_y = \int_A y \, dA \)

### Centroid of Composite Plane Laminas:

\( \bar{x} = \frac{Q_x}{A} \)

\( \bar{y} = \frac{Q_y}{A} \)

## 2.1 Equilibrium of System of Coplanar Forces:

### Conditions of Equilibrium:

**Concurrent Forces:** \(\sum \mathbf{F} = 0\) and \(\sum \mathbf{M} = 0\) (Net force and net moment are zero)

**Parallel Forces:** \(\sum F = 0\) and \(\sum M = 0\) (Net force and net moment are zero)

**Non-concurrent Non-parallel Forces:** \(\sum \mathbf{F} = 0\) and \(\sum \mathbf{M} = 0\) (Net force and net moment are zero)

**Couples:** \(\sum \mathbf{M} = 0\) (Net moment is zero)

### Equilibrium of Rigid Bodies:

Free Body Diagrams:

- Identify and isolate the body
- Show all external forces and couples
- Apply conditions of equilibrium

## 2.2 Equilibrium of Beams:

### Types of Beams:

**Simple Beams:** Supported at both ends

**Compound Beams:** Combinations of simple beams

### Types of Supports and Reactions:

**Supports:**

- Hinged Support (Pin)
- Roller Support
- Fixed Support

**Reactions:**

- Vertical Reaction (\(R_V\))
- Horizontal Reaction (\(R_H\))
- Moment Reaction (\(M\))

### Determination of Reactions:

For various types of loads on beams, use the equations:

- Sum of Vertical Forces: \(\sum F_V = 0\)
- Sum of Horizontal Forces: \(\sum F_H = 0\)
- Sum of Moments: \(\sum M = 0\)

<!DOCTYPE html> Friction and Kinematics Formulas

## 03 Friction:

### Static Friction:

Force of static friction (\(F_{\text{static}}\)):

\[ F_{\text{static}} \leq \mu_s \cdot N \]

### Dynamic/Kinetic Friction:

Force of kinetic friction (\(F_{\text{kinetic}}\)):

\[ F_{\text{kinetic}} = \mu_k \cdot N \]

### Coefficient of Friction:

Static friction coefficient (\(\mu_s\))

Kinetic friction coefficient (\(\mu_k\))

### Angle of Friction:

Angle of friction (\(\theta\)):

\[ \tan \theta = \frac{{\text{Opposite side}}}{{\text{Adjacent side}}} = \frac{{\mu_s}}{{1}} \]

### Laws of Friction:

- Friction is proportional to the normal force.
- Friction is independent of the apparent area of contact.
- Friction is independent of the sliding velocity.
- Friction depends on the nature of the surfaces in contact.

### Concept of Cone of Friction:

The cone of friction represents all possible directions of the frictional force.

### Equilibrium of Bodies on Inclined Plane:

Forces along the inclined plane:

\[ F_{\text{parallel}} = W \cdot \sin \theta \] \[ F_{\text{perpendicular}} = W \cdot \cos \theta \]

Equations for equilibrium:

\[ \sum F_x = 0 \] \[ \sum F_y = 0 \] \[ \sum M = 0 \]

Application to problems involving wedges and ladders.

## 04 Kinematics of Particle:

### Motion of Particle with Variable Acceleration:

Acceleration (\(a\)) as a function of time:

\[ a = \frac{dv}{dt} \]

### General Curvilinear Motion:

Components of acceleration:

\[ a_t = \frac{dv}{dt} \] \[ a_n = \frac{v^2}{r} \]

### Tangential & Normal Component of Acceleration:

Tangential component (\(a_t\)) and normal component (\(a_n\)) of acceleration.

### Motion Curves:

Acceleration-time (\(a-t\), velocity-time (\(v-t\), and displacement-time (\(s-t\) curves.

### Application of Concepts of Projectile Motion:

Related numerical formulas.