Q1) Explain the concept of force and force system. Classify force system. Also explain the concept of Resultant of a force system with proper examples and sketches.

Ans: A force is a physical quantity having magnitude as well as direction. Therefore, it is a vector quantity. It is defined as an "external agency" which produces or tends to produce or destroys or tends to destroy the motion when applied on a body.

Its unit is Newton (N) in S.I. systems and dyne in C.G.S. system.

When two or more forces act on a body or particle, it is called force system. Therefore, a force system is a collection of two or more forces.

Force System Classification:

There are mainly seven types of system of forces.

1. Coplanar forces

2. Collinear forces

3. Concurrent forces

4. Non coplanar forces

5. Non collinear forces

6. Non concurrent forces

7. Parallel forces

Coplanar force system:

If all the forces of a force system are acting in the same plane, then the force system is called as coplanar force system. A coplanar force system may be concurrent, non-concurrent or parallel.

Non coplanar force system:

If forces of a force system act at different planes, they are called non coplanar force system.

Collinear force system:

The forces which are acting along a straight line are called as collinear forces.

Non collinear force system:

The forces which are not acting along a straight line are called as non collinear forces.

Concurrent force system:

If the line of actions of two or more forces passes through a certain point simultaneously then they are called concurrent forces. Concurrent forces may or may not be coplanar.

Non concurrent force system:

If the forces of a force system are such that their lines of action do not intersect at a common point, they are called non concurrent force system.

Parallel force system:

If the line of actions of all the forces of a force system is parallel to each other, they are called as Parallel forces or parallel force system.

A parallel force system may be of two types

• Like parallel forces or force system

• Unlike parallel forces or force system

Q2) What are the different methods of Force Addition? Explain them with proper sketches and geometrical derivation.

Ans: Any two forces can be added by the principle of vector addition. Two forces P and Q can be added by either by

• Triangle’s law of vector addition

• Parallelogram law of vector addition

For more than two forces, they can be added by

• Polygon law of vector addition

• Force resolution method

Triangle's law of force addition:

When two concurrent forces are in order, they are added by triangle's law of force addition. Let two concurrent forces act on a particle as shown in the figure. They can be represented by the two sides of a triangle in order. Then the third side of the triangle opposite the order will represent resultant of both the forces. The resultant is generally denoted by R.

Parallelogram law of force addition:

It states “If two forces simultaneously acting at a point be represented in magnitude and direction by two adjacent sides of a parallelogram, the diagonal will represent resultant in magnitude and direction, passing through the point of intersection of two forces.

Polygon’s law of force addition:

The polygon’s law of force addition states "If a number of force acting at a point are such that they can be represented in magnitude and direction by the sides of a open polygon taken in order, then their resultant is represented in magnitude and direction by the closing side of the polygon but taken in the opposite order".

Force resolution method of force addition:

If a force system has n number of forces, then the resultant can be found by determining the horizontal components and vertical components of all the forces.

If the sum of all the horizontal components be ∑Fxi and the sum of all the vertical components be ∑Fyi , then the resultant force can be determined by

R = √(∑Fxi² + ∑Fyi²)

Q3) Explain the concepts of Equilibrium and Supports. What are the Supports Reactions? How can we calculate Support Reactions for a force system in Equilibrium.

Ans: EQUILIBRIUM:

A body is said to be in static equilibrium when there is no change in position as well as no rotation exist on the body. So to be in equilibrium process, there must not be any kind of motions ie there must not be any kind of translational motion as well as rotational motion.

SUPPORTS and SUPPORT REACTION:

The restriction to the motion of a body in any direction is called a constraint or supports. When a ball is resting on a smooth surface horizontal movement is possible, but vertical downward movement is restricted. So, according to the Newton’s third law of motion, there will be a normal reaction acting opposite to the weight of the ball which will be equal to the magnitude of the weight.

In general, the action of a constraint body on any support induces an equal and opposite reaction from the support.

We can determine the value of unknown reactions from the equation we get applying the conditions of equilibrium to the free body diagram of the object at equilibrium under the action of a force system and support reactions.

For Concurrent force system, the equilibrium conditions are as follows

(i) ΣFx= 0; (ii) ΣFy = 0

Suppose a ball at rest on the ground as shown in the figure. Here the support reaction is unknown and the value of the reaction can be obtained from the equation of equilibrium. Here, ΣFx= 0 and we can write,

R + W = 0;

If the W = 50 N, then R – 50 = 0 (taking downward force as negative)

o R = 50 N

Q4) What is Equilibrant and what is its relation with resultant force?

Ans: Resultant:

The resultant of a force system is the Force which produces same effect as the combined forces of the force system would do. So if we replace all components of the force by the resultant force, then there will be no change in effect.

The Resultant of a force system is a vector addition of all the components of the force system. The magnitude as well as direction of a resultant can be measured through analytical method.

Equilibrant:

Any concurrent set of forces, not in equilibrium, can be put into a state of equilibrium by a single force. This force is called the Equilibrant. It is equal in magnitude, opposite in sense and co-linear with the resultant. When this force is added to the force system, the sum of all of the forces is equal to zero.

Q5) Explain Lami's theorem with proper sketches & geometrical proof. How can we find the resultant force of a force system? Explain the procedure.

Ans: Lami’s theorem deals with the equilibrium condition of a body under three concurrent forces.

“It states that, if a body is in equilibrium under the action of three concurrent forces, each force is proportional to the “sine” of the angle between the other two forces.”

Here, three forces F1, F2, F3 are in equilibrium and α, β, γ are the corresponding angles as shown in the figure. Then according to Lami’s theorem,

(F1/sin α) = (F2/sin β) = (F3/sin γ)

Proof of the Lami’s theorem:

Let P, Q, R be the three forces acting at a point O in equilibrium as shown in the figure. Let R1 be the resultant of two forces P and Q. Now point O is subjected to only two forces R and R1. Therefore, at equilibrium |R| = |R1|. R and R1 must be equal, opposite and collinear.

As AO is parallel and equal to BC, hence BC will be equal to Q.

Now, in ∆OBC,

∠BOC = π ‒ β

∠AOC = π ‒ α

∠BCO = π ‒ (π ‒ β) ‒ (π ‒ α) = α + β ‒ π

But, α + β + γ = 2π

α + β = 2π ‒ γ

∠BCO = α + β ‒ π

∠BCO = 2π ‒ γ ‒ π

∠BCO = π ‒ γ

Now from the properties of triangle,

P/sin ( π ‒ α) = Q/sin (π ‒ β) = R/sin (π ‒ γ)

= (P/sin α) = (Q/sin β) = (R/sin γ)

Q6) Compare the concept of equilibrium for coplanar concurrent as well as coplanar non-concurrent force system. Explain the physical quantity Moment of a force.

Q7) State and prove "Varignon's Theorem of Moments."

Q8) Explain the concept of Resultant Moment. How can we find the position of a resultant force in the analysis of force system.

Q9) If three forces of equal magnitudes act along the three sides of an Equilateral Triangle, can we achieve Static Equilibrium?

Solution: If three forces of equal magnitude act along the sides of an equilateral triangle in order, the net force will zero, but the net moment is not zero, so equilibrium can not be achieved.

Q10) What is a Couple? What are the properties of a couple? Differentiate between moment and couple.

Ans: Two unlike parallel, non-collinear forces having same magnitude from a Couple. The distance between two forces is known as arm or lever of the couple.

Properties of a Couple:

(i) Two unlike parallel, non-collinear forces having same magnitude from a Couple.

(ii) The resultant of a couple is always zero.

(iii) The moment of a couple is the product of one of the forces and moment arm of the couple.

(iv) A couple cannot be balanced by a single force.

(v) It can be balanced only by another couple of opposite nature.

(vi) The moment of a couple is independent of the moment center.

(vii) The effect of a couple is unchanged if

(a) it is shifted to any other position in its plane or to any other parallel plane

(b) it is rotated through any angle in its plane.

Q11) What is a coplanar concurrent force system?

Ans: If all the forces of a force system are such that all of them are acting on a single plane and the line of actions of all the forces intersect at a common point on the plane, then they are called Coplanar concurrent force system.

Q12) What is coplanar non concurrent force ...