Module 1: Resultant

 COPLANAR FORCE SYSTEM

        1. Coplanar Collinear Force System:

If two or more forces acting along the same line (or the line of action is same) lying on the same plane then the force system is called as coplanar collinear force system.

 

Direction of the forces does not matter only the condition is that all the forces must act along the same line (or the line of action of all the forces must be same). The line of action may be horizontal, vertical or inclined. 

 

Following figure shows force F1 acting toward left and forces F2, F3 and F4 acting towards right along the same line (the line of action is same) and all the forces lying on the same plane therefore this force system is called as coplanar collinear force system.

      2. Coplanar non-collinear Force System:

If two or more forces acting along the same line (or the line of action is same) lying on the same plane then the force system is called as coplanar collinear force system.

Direction of the forces does not matter only the condition is that all the forces must act along the same line (or the line of action of all the forces must be same). The line of action may be horizontal, vertical, or inclined.  

Following figure shows force F1, F2, and F3 not acting along the same line ( or the line of action is not same) but all the forces lying on the same plane therefore this force system is called as coplanar non-collinear force system.

      3. Coplanar concurrent force system:

If all the forces (or system of the forces) passing through the same point (or meet at the same point) and lying on the same plane then the force system is called as coplanar concurrent force system.

Direction of the forces does not matter only the condition is that all the forces must pass through the same point or meet at the same point. Nature of forces can be push or pull.

Following figure shows force F1, F2, F3, F4, and F5 passing through the same point and lying on the same plane therefore this force system is called as coplanar concurrent force system.

      4. Coplanar non-concurrent force system:

If all the forces (or system of the forces) does not pass through the same point (or not meet at the same point) and lying on the same plane then the force system is called as coplanar non-concurrent force system.

            Following figure shows force F1, F2, F3, F4, and F5 not passing through the same point and                    lying on the same plane therefore this force system is called as coplanar non-concurrent force                  system. 

 

        5. Coplanar parallel force system:

               There are two types of coplanar parallel force system,

                I) Coplanar Like parallel force system

               II) Coplanar unlike parallel force system

 

                I) Coplanar Like parallel force system:

                   If all the forces are parallel with each other and their arrow heads pointing in the same                             direction & lying     on the same plane such force system is called as coplanar Like parallel                     force system.

              II) Coplanar unlike parallel force system:

                       If all the forces are parallel with each other and their arrow heads pointing in the                                       different direction & lying on the same plane such force system is called as coplanar                               unlike parallel force system.

 

     6. Coplanar non-concurrent non- parallel force system 

         (general force system):

             If all the forces are not passing through the same point and not parallel with each other such a                 force system is called as coplanar non-concurrent non- parallel force system (general force                     system).

NON-COPLANAR FORCE SYSTEM

1. Non-coplanar concurrent force system:

If all the forces pass through the same point but lying on the different planes such a force system is called a non-coplanar concurrent force system.

In the following figure all forces i.e. force F1, F2, F3 and F4 passing through the same point and lying on the different planes (force F3 is common for both the planes on which force F2 and F3 lying) therefore this force system is called as non-coplanar concurrent force system.

2. Non-coplanar non-concurrent force system:

If all the forces are not passing through the same point and lying on the different planes such a force system is called as non-coplanar non-concurrent force system.

In the following figure all forces i.e. force F1, F2, F3 and F4 are not passing through the same point and lying on the different planes therefore this force system is called as non-coplanar non-concurrent force system.

3. Non-coplanar parallel force system:

If all the forces are parallel to each other and lying on the different planes such a force system is called a non-coplanar parallel force system.

This force system has following two types,

i)  Non-coplanar like parallel force system

ii) Non-coplanar unlike parallel force system

i)  Non-coplanar like parallel force system:

If all the forces are parallel to each other, acting in the same direction and lying on the different planes such a force system is called a non-coplanar like parallel force system. 

In the following figure all forces i.e. force F1, F2, and F3 are parallel to each other and lying on the different planes therefore this force system is called as non-coplanar like parallel force system.

ii) Non-coplanar unlike parallel force system:

If all the forces are parallel to each other, acting in different directions and lying on the different planes such a force system is called as non-coplanar unlike parallel force system.

In the following figure all forces i.e. force F1, F2, and F3 are parallel to each other, acting in different directions (force F1 and F2 acting downward and force F3 acting upward) and lying on the different planes therefore this force system is called as non-coplanar unlike parallel force system.

4. Non-coplanar non-concurrent non-parallel force system:

If all the forces not passing through the same point, not parallel to each other and lying on different planes such a force system is called as non-coplanar non-concurrent non-parallel force system.

In the following figure all forces i.e. force F1, F2, and F3 are not passing through the same point, not parallel to each other and lying on the different plane such force system is called as Non-coplanar non-concurrent non-parallel force system.

RESOLUTION OF FORCES

This is the method of representing a single force into the number of forces without changing the effect of the force on the body.

Suppose a single force F is resolved into the number of forces F1, F2, F3, etc. these resolved forces are called as components of the single force F. And this force F is called as resultant of the resolved forces.

If a force F is resolved into two axes at right angles to each other then each force is called as resolved part of a force or components of a force.

If a single force F is resolved along two directions which are not perpendicular to each other then each force is called as component of a force (not resolved part of a force). 

Methods of resolution:

There are two methods of resolution,

i) Resolution of a force into two mutually perpendicular components (or directions)

ii) Resolution of a force into two non-perpendicular components (or directions)

COMPOSITION OF FORCES

Composition (or compounding) is the procedure to find out sing resultant force of a force system.

There are two methods of composition of forces.

1. Analytical method

2. Graphical method

1. Analytical Method:

i) Resultant of coplanar collinear forces:

   R = F1 + F2 - F3 + F4

Here addition of all the forces represent the magnitude of resultant force and its sign (i.e. '+' or '-') represent the direction of action of resultant force.

ii) Resultant of coplanar concurrent forces:

a) Parallelogram law of forces (trigonometric method):

It is used to find out the resultant of only two concurrent forces.

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

Here, Q =  OB = AC and OB parallel to AC

OA = P and OC = R

AD = Q cosѲ and CD = Q sinѲ

In right angle triangle  ∆ODC

(OC)² = (OD)² + (DC)²

(OC)² = (OA + AD)² + (DC)²

R² = (P + QcosѲ)² + (QsinѲ)²

R² = P² + Q²cos²Ѳ + ²PQcosѲ + Q²sin²Ѳ

R² = P² + Q² (cos²Ѳ + sin²Ѳ) + ²PQcosѲ

R² = P² + Q² + 2PQcosѲ    ………………  (cos²Ѳ + sin²Ѳ = 1)

R = √ (P² + Q² + 2PQcosѲ)   ……………..(magnitude of resultant force)

Where, Ѳ is the angle between the forces P and Q.

If P and Q are known the magnitude of the resultant R can be found out.

Let, R make an angle of α with the direction of P.

In right angle triangle  ∆ODC,

Tanα = CD / OD = CD / (OA + AD) = QsinѲ / (P + Q cosѲ)

α = tan-1 (QsinѲ / (P + Q cosѲ)) ……………………………(direction of resultant force)

If, P, Q and Ѳ are known direction of the resultant can be determined.

b) Method of resolution (Algebraic method):

This method is used to find out the resultant of more than two concurrent forces.

Following are the steps to find out the resultant,

i) Resolve all the forces horizontally and find the algebraic sum of all the horizontal components i.e. ∑F_x

ii) Resolve all the forces vertically and find the algebraic sum of all the vertical components i.e. ∑F_y

iii) Resultant is given by,

R = √ [(∑F_x)² + (∑F_y)²]

Let, ϴ be the acute angle (less than 900) made by R with horizontal.

Then, tanϴ = ∑F_y / ∑F_x

ϴ = tan^-1 (∑F_y / ∑F_x)

Varignon’s theorem:

The algebraic sum of moments of all the forces about a point is equal to the moment of the resultant force about the same point.

∑M_FA = M_RA

 

Resultant of coplanar parallel forces:

i) Magnitude of the resultant is equal to the algebraic sum of all the forces i.e.

R = ∑F

ii) direction of the resultant can be determined from the sign of the resultant (positive or negative)

iii) position of the resultant can be located by using Varignon’s theorem of moments

∑M_FA = M_RA

Question 1:  A system of parallel, non-concurrent forces is acting on a rigid bar. Reduce the system of forces to 

I) A single force R & its position 

II) A single force R & a couple at B

Solution:

This is coplanar unlike parallel force system. In this force system the magnitude of the resultant can be calculated by the sum of all the parallel forces. The sign of the sum i.e., ‘ + ’ or ‘ – ‘ represent the direction of resultant. ‘ + ‘ sign indicates direction of resultant along positive x-axis and y-axis and the ‘ – ‘ sign indicates the direction of resultant along the negative x-axis and y-axis.

I) A single force R & its position:

A single force (or resultant) is given by,

Resultant = Summation of all the parallel forces

∴   R = Σ F            

         =  + (50 N) – (200 N) – (100 N) + (70 N)

      R = - 180 N (Downward) 

Here negative sign indicates that the resultant acting downward direction.

The position of the single force can be calculated by using Varignon’s theorem,

(The summation of moment of all the forces about point ‘A’ is equal to the moment of their resultant about the same point ‘A’).

∴ Σ M_FA= M_RA

(50 N x 0 m) – (200 N x 1 m) – (100 N x 3 m) + ( 70 N x 4 m) = (- 180 N x d m)

                                                         (0 – 200 – 300 + 280) N.m = (- 180 N x d)

                                                                                    - 220 N.m = (- 180 N x d)

Here negative sign (sign of - 220 N.m) shows that the moment of resultant is in anticlockwise direction.

                                                         - 220 N.m / -180 N = d

                                                                             1.22 m = d

∴ d = 1.22 m

The position of the single force (or resultant) is 1.22 m from and at the right side of point ‘A’, shown in figure below.

 

II) A single force R & a couple at B:

To complete the given condition, shift the resultant at point ‘B’ as shown in figure below. Now at the same point ‘B’ draw the another resultant equal in magnitude and opposite in direction and collinear to the shifted resultant. There effect of these two opposite resultant will get cancelled.

Couple at point B  =  Magnitude of one of the forces x arm of the couple

                               =  180 x 0.22

Couple at point B =  39.6 N.m

Question 2: Figure shows a parallel system of four forces and two couples.

I) Replaced it by a single force and obtain its location from Point A

II) Replace it by a force couple system at Point A

III) Replace it by a force couple system at Point D

IV) Replace it by two parallel forces at B and D

 Solution:

       I) Replaced it by a single force and obtain its location from Point A:

This is a system of four parallel forces and two couples

Single force (or Resultant force) 

R = Σ F            

R = - 8 - 4 - 10 + 12

R = - 10 kN   or   R= 10 kN (downward)

Locations of resultant force

      Using Varignon’s theorem

      (8 x 0) - (4 x 3) – (10 x 6) + (12 x 9) + 30 – 80 = - (10 x d)

d = 1.4 m  (right of A)

The resultant is R = 10 kN (downward) and is located at a perpendicular distance d = 1.4 m right of A as shown in figure below.

      II) Replace it by a force couple system at Point A:

To replace it by a force couple system at point A, we need to shift resultant force R = 10 kN to A by introducing a couple M. The perpendicular distance between point A and force R is 1.4 m.

Couple M = F x d

                 = + (10 x 4.6)

Couple M = 46 kN.m

Or         M = 46 kN.m (anticlockwise)

       The resultant force couple at A is shown in figure below.

      III) Replace it by a force couple system at Point D:

To replace it by a force couple system at point D, we need to shift the force R= 10 kN to D by introducing a couple M.

The perpendicular distance d of force R is 4.6 m

Couple M = F x d

                   = (10 x 4.6)

                   = 46 kN.m

Couple M = 46 kN.m (anticlockwise)

The force couple system is shown in figure below.

           IV) To replace it by two parallel forces at B and D:

The force couple system D is shown in figure. The couple of 46 kN.m can be replaced by two parallel forces at B and D, equal in magnitude and opposite in sense.

Couple M = F x d

            46 = F x 3       (distance between B and D is 3 m)

              F = 15.33 kN

Therefore F = 15.33 kN (downward) at B and F = 15.33 kN (upward) at D can replace the couple of 46 kN.m

Adding forces at D i.e. – 10 + 15.33 = 5.33 kN,

We get the two parallel components as 15.33 kN (downward) at B and 5.33 kN (upward) at D as shown in figure below.