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.
