System-of-Coplanar-Forces

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Engineering Mechanics Mechanics

System of coplanar forces

SYSTEM OF COPLANAR FORCES INTRODUCTION The Force is an external effort in the form of push or pull, which (i) (ii) (iii)

produces or tries to produce motion in a body at rest, or stops or tries to stop a moving body, or changes or tries to change the direction of motion of the body. Force

Since mass do not change Force F

∝ rate of change of momentum ∝ rate of change of (mass × velocity) ∝ mass × rate of change of velocity ∝ mass × acceleration ∝ m×a

...(1.1)

= k ×m ×a Wher Where, e, F is the force, force , m is the t he mass ma ss and an d a is the the acce accele lera rati tion on and and k is the t he constan co nstantt of proportio propo rtionalit nality. y. In all the systems, unit of force is so selected that the constant of the proportionality becomes unity. For example, in S.I. system, unit of force is Newton, which is defined as the force that is required to move 2 one kilogram (kg) mass at an acceleration of 1 m/sec . 2 One newton = 1 kg mass × 1 m/sec Th us k =1 F = m×a .. .( 1. 2) 2 However in MKS acceleration used is one gravitational acceleration (9.81 m/sec on earth surface) and unit of force is defined as kg-wt. Thus F in kg wt = m× m×g g ...(1.3) Thus 1 kg-wt = 9.81 newton …(1.4) It may be noted that in usage kg-wt is often called as kg only.

∴

The following examples illustrate the above definition: 



When we push a ball lying on the ground, it starts rolling. The force exerted has thus produces motion in the ball. However, when we push a heavy stone, it does not move. The effort made in this case has only tried to produce motion, but has not succeeded. When a piece of stone tied to one end of a string is whirled in a circle, a











Characteristics Characteristics of a force:To define force we need four characteristics characteristics which are as under   

Magnitude of the force (ex- 10kg or 5N, 15 tonnes) Direction of line along which force is acts (ex- towards E,W,N,S) Nature of the force (ex- Push or Pull) Point at which the force acts.

SYSTEM OF FORCES When two or more forces act on a body, they are called to for a system of forces. Following systems of forces are important from the subject point of view:

1. Coplanar forces: The forces, whose lines of action lie on the same plane, are known as coplanar forces. 2. Collinear forces: The forces, whose lines of action lie on the same line, are known as collinear forces. 3. Concurrent forces: The forces, which meet at one point, are known as concurrent forces. The concurrent forces may or may not b collinear. 4. Coplanar concurrent forces: The forces, which meet at one point and their line of action also lay on the same plane, are known as coplanar concurrent forces. 5. Coplanar non-concurrent forces: The forces, which do not meet at one point, but their lines of action lie on the same, are known as coplanar non-concurrent forces. 6. Non-Coplanar concurrent concur rent forces: The forces, which meet at one point, but their lines of action do not lie on the same plane, are known as non-coplanar concurrent forces. 7. Non-Coplanar non-concurrent forces: The forces, which do not meet at one point and their lines of action do not lie on the same plane, are called non-coplanar non-concurrent forces.

TYPE OF FORCE SYSTEM

POSSIBLE RESULTANT

Concurrent

Force

Coplanar, Non-Concurrent

Force or a couple

Parallel, Non-Coplanar, Non-Concurrent

Force or a couple

Non-parallel, Non-Coplanar, Non-Concurrent

Force or a couple











RESULTANT FORCE If a number of forces, P,Q,R ….etc, ….etc, are acting simultaneously on a particle, then it is possible to find out a single force which could replace them i.e. which would produce the same effect as produces by all the given force. This single force is called resultant force and the given forces P, Q, R … etc are called component forces.

Methods For The Resultant Force Though there are many methods for finding out the result force of a number of given forces, yet the following are important from the subject point of view. 1.

Analytical method

2. Methods of resolution

Analytical Method For Resultant Force The resultant force of a given system of force may be found out analytically by the following methods. 1.

Parallelogram law of forces

2.

Graphical method

Parallelogram Law of Forces It state “If two forces, acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram; their resultant may be represented in magnitude and direction by the diagonal of the parallelogram, which passes through their point of intersection.” Mathematically, resultant force.

          Where P and Q =

=

Angle between the forces P and Q, and

α=

Angle which the resultant force makes with one for the forces (say P)

Note: If the angle

Then

Forces whose resultant is required to be found out

() which the resultant force makes with other force    

Q,











Cor.

   )            (since   ) If  when the forces act along the same straight line but in opposite direction, then (since  ) 

i.e., when the forces act along the same, then 1. If R=P+Q (since 2. If i.e., when the forces act at right angle, then

3.

i.e.,

In this case, the resultant force will act in the direction of the greater force. 4. If the two force are equal i.e., when P = Q, then



=    √       ( )                      

Example.1 Two forces of 100 N and 150 N are acting simultaneously at a point. What is the resultant of these two forces, if the angle between them is 45°? Solution. Given: P = 100N ; Q = 150 N and We know that resultant of the two forces,



             ( ))      

N

N

= 250 N (Ans). Example 1.2. Two forces act at an angle of 120°. The bigger force is of 40 N and the resultant is perpendicular to the smaller one. Find the smaller force.











           From the geometry figure, we find that   

Solution Given:

Let

Q = Smaller force in N

We know that

           ()                              N(Ans). General Laws for the Resultant Force The resultant force, of a given system of forces, may also be found out by the following general laws: 1.

Triangle law of force

2.

Polygon law of forces.

Triangle Law of Forces It states, “If two forces acting simultaneously on a particle be represented in magnitude and direction by the two sides of a triangle, taken in order; their resultant may be represented in magnitude and direction by the third side of the triangle, taken in opposite op posite order.”













Polygon Law of Forces It is an extension of Triangle Law of Forces for more than two force which states, “If a number of forces acting simultaneously on a particle, be represented in magnitude and direction, by the sides of a polygon taken in order; then the resultant of all these forces may be represented, in magnitude and direction, by the closing side of the polygon, taken in opposite order.

Graphical (Vector) Method for the Resultant Force It is another name for finding out the magnitude and direction of the resultant force by the Polygon law of forces. It is done as discussed below: 1. Construction of space diagram (position diagram)- It means the construction of a diagram showing the various forces (or loads) along with their magnitude and lines of action. 2. Use of Bow’s notations- All the forces in the space diagram are named by using the Bow’s notations. It is a convenient method in which every force (or load) is named b y two capital letters, placed on its either side in the space diagram. 3. Construction of vector diagram (force diagram)- It means the construction of a diagram











Now draw the vector diagram for the given system of forces as shown in (b) and as discussed below. 1. Select some suitable point a and draw ab equal to 50 N to some suitable scale and parallel to the 50 N force of the space diagram. 2. Through b, draw bc equal to 100 N to the scale and parallel to the 100 N force of the space diagram. 3. Similarly through c, draw cd equal to 130 N to the scale and parallel to the 130 N force to the space diagram. 4. Join ad, which gives the magnitude as well as direction of the resultant force. 5. By measurement, we find the magnitude of the resultant force is equal to 70 N and acting











RESULTANT OF CONCURRENT FORCES Resultant of a force system is a force or a couple that will have the same effect to the body, both Resultant of in translation and rotation, if all the forces are removed and replaced by the resultant.

The equations involving the resultant of force system are the following 1.

The x-component of the resultant is equal to the summation of forces in the x-direction.

2. The y-component of the resultant is equal to the summation of forces in the y-direction. 3.

The z-component of the resultant is equal to the summation of forces in the z-direction. Note that according to the type of force system, one or two or three of the equations above will be used in finding the resultant

CONCURRENT FORCE SYSTEM Concurrent force system: The resultant of a concurrent forces system can be defined as the simplest single force which can replace the original system without changing its external effect on a rigid body A concurrent force system contain force whose whos e line-of-action meet at some one point.

Force may be Tensile (Pulling) as well as Compressive (Pushing)











PARALLEL FORCE SYSTEM Parallel forces can be in the same or in opposite directions. The sign of the direction can be chosen arbitrarily, meaning, taking one direction as positive makes the opposite direction negative. The complete definition of the resultant is according to its magnitude, direction, and line of action.

Non-Concurrent force system: The resultant will not necessarily be a single force but a force system comprising a force or a couple or a force and a couple. The type of force system as classified below along with their possible

The resultant of non-concurrent force system is defined according to magnitude, inclination, and position. The magnitude of the resultant can be found as follows











Where, Fx

=

component of forces in the x-direction

Fy

=

component of forces in the y-direction

Rx

=

component of the resultant in x-direction

Ry

=

component of the resultant in y-direction

R

=

magnitude of the resultant

θx

=

angle made by a force from the x-axis

MO

=

moment of forces about any point O

d

=

moment arm

MR

=

moment at a point due to resultant force

ix

=

x-intercept of the resultant R

iy

=

y-intercept of the resultant R

NON-CONCURRENT NON-PARALLEL The principles of equilibrium are also used to determine the resultant of non-concurrent, non-













Moment of a Force Above, we discussed about the effects of forces, acting on a body, through their lines of action or at the point of their intersection. But in Moment of Force , the effect of these forces, at some other point, away from the point intersection on their lines of action. It is the turning effect produced by a force, on the body, on which it acts. The moment of a force is equal to the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force. Mathematically, moment

Where

   Force acting on the body, and











VARIGNON’S THEOREM It states that the moment of a force about any point is equal to the sum of the moments of its components about the same point. This principle is also known as principle of moments. Varignon’s theorem need not be restricted to the case of only two components but applies equally well to a system of forces and its resultant. For this it can be slightly modified as, “the algebraic sum of the moments of a given system of forces about a point is equal to the moment of their resultant about the same point This principle of moment may be extended to any force system. Proof : Referring to the fig Let R be the resultant of forces F 1 and F2 and B the moment centre.

Let d, d1 and d2 be the moment arms of the forces, R, F1 and F2 respectively from the moment centre B. Then in this case, we have to prove that Rd

= F1d1+F2d2

Join AB and consider it as y axis and draw x axis at right angles to it at A (fig-below). Denoting by the angle that R makes with x axis and nothing that the same angle is formed by perpendicular to R at B with Ab 1. We can write;



Rd

 = AB × (R  )

= R × AB










F1d2, F2d2


= AB (F1x+F2x) …. (d)

= AB Rx From, equation (a) and (d) we get Rd = F1d1+F2d2

If a system of forces consists of more than two forces, the above result can be extended as given below: Let F1, F2, F3 and F4 be four concurrent forces and R be their resultant. Let d1, d2, d3, d4 and a be the distance of line of action of force F 1, F2, F3 and F4 and R1 respectively from the moment centre O. (ref fig - 2.7) If R1 is the resultant of R 1 of F1, F2, and its distance O is a 1, then applying Varignon’s theorem. F1a1 = F1d1+F2d2 If R2 is the resultant of R 1and F1, (and hence of F1, F 2, F3) and its distance from O is a 2, then applying Varignon’s theorem. R2a2

= R1a1+F3d3 =F1d1+F2d2+F3d3










Example


Find the moment of 100 N force acting at B about point A as shown in fig 2.8



Solution 100 N force may be resolved into its horizontal components as 100 and vertical component 100 . From Varignon’s theorem moment of 100 N force about the point A is equal to sum of the moment of its components about A .



Taking clockwise moment as positive,

    











        By varignon’s theorem, moment of 5000 N force about A is equal to moment of its component force about the same point.

      Couple’s A pair of two equal and unlike parallel forces (i.e. forces equal in magnitude, with lines of action parallel to each other and acting in opposite directions) is known as a couple.

Arm of a Couple

The Perpendicular distance (a), between the lines of action of the two equal and opposite parallel












1. Clockwise couple,


2.

Anticlockwise couple

It is the moment of a force, whose effect is to turn or rotate the body, in the same direction in which hands of a clock move as shown in fig.

2. It is the moment of a force, whose effect is to turn or rotate the body, in the opposite direction in which the hands of a clock move as shown in the fig.











Units of Moment of Inertia

The unit of moment of inertia of a plane area depends upon the units of the area and the length. e.g. 2

4

1. If the area is in m and the length is also in m, the moment of inertia expressed in m . 2

4

2. If area is in cm and the length is also in cm, then moment of inertia is expressed in cm . 2

3. If the area in mm and the length is also in mm, then moment of inertia is expressed in 4 mm . Theorem of Moment of Inertia

1. Perpendicular axis

System-of-Coplanar-Forces

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