Table of Contents

#### CONTENTS

**System of Forces****Equilibrium Equations****Internal Forces in Structures****Principle of Transmissibility****Varignon’s Theorem****Couple****Friction****Self Locking Mechanism****Kinematics****Centre of Mass****Centroid of Different Figures****Euler’s Equation of Motion****Impulse-Momentum****Conservation of Linear Momentum****Angular Impulse and Angular Momentum****Energy Methods****Potential Energy**

#### SYSTEM OF FORCES

Force is the action of one body on another. A force tends tomove a body in the direction of its action. The action of a force is characterized by its magnitude,by the direction of its action, and by its point of application. Thus, force isa vector quantity.

#### Different Types of Force Systems

**Collinear Forces **– In this system, line of action of all the forces act along the same line

**Coplanar parallel force:** In this system, all forces are parallel to each other and lie in a single plane.

**Coplanar concurrent point**: In this system, line of action of all forces passes through a single point and forces lie in the same plane.

**Coplanar non-concurrent forces **– All forces do not meet at a point but lie in a single plane.

**Non-coplanar parallel forces –** In this system all the forces are parallel to each other but not in the same plane.

**Non-coplanar concurrent forces –**In this system, all forces do not lie in the same plane but their line of action passes through a single point.

One system is in which, all forces do not lie in the same plane and their lines of action do not pass through a single point; they may not be parallel.

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#### Free-Body Diagrams

In applying the principles of mechanics to analyse forces acting on a body, it is essential to isolate the body in question from all other bodies so that a complete and accurate amount of all forces acting on this body can be taken. The diagram of such an isolated body with the representation of all external forces acting on it is called a free-body diagram.

**Three Force Theorem –** If three forces are acting on a body and then the system of three forces is in equilibrium, must be coplanar, and either be concurrent or be parallel – that is a three force theorem.

#### Lami ’s Theorem

When three coplanar and concurrent forces acting on a body are kept in equilibrium, then each force is proportional to the sine of the angle between the other two and the constant of proportionality is the same

The forces acting on a body are of two types:

**External Forces**: These forces on structures are the stresses that act on a structure from outside. For example: gravity force, impact force**Internal Forces:**These forces act between different parts of the same structure. Internal forces are of four types: tension, compression, torsion and shear.

**Tension Force** – The force which causes all of the particles of an object to pull apart is called tension.

**Compression Force **– The force that presses or squeezes the particles of an object together is known as compression.

#### Torsion Force

The internal twisting forces created in an object as aresult of a twisting motion being applied to the object is known as torsion.

#### Shear Force

The forces acting in an object as a result of pushes and/or pulls in opposite directions; usually results in rips or tears in an object is known shear.

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#### Principle of Transmissibility

The principle of transmissibility states that a force may be applied at any point on its given line of action without altering the resultant effects of the force external to the rigid body on which it acts. For example: The two forces F and F’ have the same effect on the rigid body and are said to be equivalent.

#### Varignon’s Theorem

This theorem states that moment of a force about any pointis equal to the sum of the moments of the components of that force about the samepoint i.e. the moment about a given point O of the resultant of severalconcurrent forces is equal to the sum of the moments of the particular forcesabout the same point

r x F_{1} + r x F_{2} + …. = r x (F_{1} + F_{2} + ….)

#### Couple

The moment produced by two equal, opposite and noncollinearforces is called a couple. For example:If two equal and opposite forces F and – F are d distance – F+F O ad apart as shown in Fig. Thesetwo forces cannot be combined into asingle forces because there sum in every direction is zero. Their only effect is to produce a tendency ofrotation. The combined moment of the towforces about an axis normal to their plane isthe couple M.

M = F(a+d) – Fa = Fd

#### Equivalent Couples

The two couples producing the same moment M areequivalent.

- The two couples are lie in the parallel plane
- The two couples have the same sense or the tendency to cause rotation in the same direction. A couple is not affected if the forces act in a different but parallel plane, as shown in Figure

#### Force-Couple System

Any force acting on a rigid body may be moved to an arbitrary point O, provided that a couple is added, of moment equal to the moment of F about O and the combination obtained is referred to as a force-couple system.

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#### FRICTION

When a body slides over another body, a resisting force is exerted at the surface of contact by stationary body on the moving body, this is called the frictional force.

#### Characteristics of Friction

Considering a block resting on a horizontal plane withirregularities at the mating surface, this rough and irregular surface wouldcause degree of unlocking and wedgingaction between the mating surface. Whentractive force P is applied to the block in parallel to contact surface, a resisting force F would begenerated automatically opposite toapplied direction of P.

#### Following are the features of frictional (Figure):

- During static condition, i.e. block is at rest,with increase in applied force P, F will increase showing the relationship ofdirect proportionality. This can beshown in region 0-1, known as zone ofstatic friction (Fig.) and point 1 is known as mass static friction.
- When block is set into motion and frictionalforce cannot be increased further, atthat situation F is called as limiting friction. This is maximum friction force at time ofimpending motion (motion is about tobegin).
- After certain point, friction force startsdropping to lower value called kineticfriction. Zone 2-3 is zone of kinetic friction (Fig.)

Hence, the main characteristics of friction force can besummarized as:

- Always acts in the direction opposite to applied motion.
- By nature it is passive force and will sustain as long as tractive force will be in action.
- Self adjusting force.

#### Nature of Friction

- Dry Friction (Coloumb friction): This comes into action when contact surfaces are dry. This is further divided into two parts: Sliding force and Rolling force.
- Fluid Friction: This comes into action when lubricating fluid is intended between contact surfaces.
- Static and Dynamic Friction Static friction develops when no relative motion exists between contacting surfaces Dynamic friction develops when there is relative motion between contacting surfaces.

#### Laws of Solid Friction

- Friction acts tangential to contact surfaces andalways in opposite direction of applied motion.
- Friction force is maximum at the time motion isabout to begin and vary from zero to maximum.
- Limiting friction is independent of area andshape of contact surfaces.
- Limiting friction depends upon nature of contactsurfaces (being rough or smooth)

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