Forces Acting at a Point And Methods of Force Resolution

Forces Acting at a Point And Methods of Force Resolution

Introduction

A force or a system of forces cannot be seen; only the effect of a force or a system of forces can be seen. That is:
● A force can change or try to change the shape of an object.
● A force can move or try to move a body that is at rest. If the magnitude of the force is sufficiently great it will cause the body to move in the direction of the application of the force. If the force is of insufficient magnitude to overcome the resistance to movement it will still try to move the body, albeit unsuccessfully.
● A force can change or try to change the motion of a body that is already moving
● The effect of any force depends on the following:
     1. The magnitude (size) of the force measured in newtons (N).
     2. The direction of the force.
     3. The point of application of the force.
     4. The ability of a body to resist the effects of the force.
● As already stated a force cannot be seen; however, it can be represented by a vector

Force is a vector quantity and thus has both a magnitude and a direction.  A vector may be represented by a straight line, the length of line being directly proportional to the magnitude of the quantity and the direction of the line being in the same direction as the line of action of the quantity. An arrow is used to denote the sense of the vector.figure 1 represents a force of 5 newtons acting in a direction due east.

FIG 1

The arrow is positioned at the end of the vector and this position is called the ‘nose’ of the vector

1- Resultant forces

The Resultant of Two Coplanar Forces
It is possible to replace two or more other forces and produce the same effect a resultant force
Since both the forces F1 and F2 have a common line of action and act in the same direction on the point P along the same line of action can be replaced by a single force having the same effect, their vector lengths can simply be added together to obtain the magnitude of the vector of the single force that can replace them This is the force FR shown in Fig. 2. It is called the resultant force

FIG 2

Note that the vector for a resultant force has a double arrowhead to distinguish it from the other forces acting in the system.
the forces been acting in opposite directions along the same line of action, then the resultant force (FR) could have been determined by simply subtracting the vector length for F1 from the vector length for F2

FIG 3

1-1 Triangle of Forces Method
A simple procedure for the triangle of forces method of vector addition is as follows:
(i) Draw a vector representing one of the forces, using an appropriate scale and in the direction of its line of action.
(ii) From the nose of this vector and using the same scale, draw a vector representing the second force in the direction of its line of action.
(iii) The resultant vector is represented in both magnitude and direction by the vector drawn from the tail of the first vector to the nose of the second vector.

FIG 3

1-2 The Parallelogram of Forces Method
A simple procedure for the parallelogram of forces method of vector addition is as follows:
(i) Draw a vector representing one of the forces, using an appropriate scale and in the direction of its line of action.
(ii) From the tail of this vector and using the same scale draw a vector representing the second force in the direction of its line of action.
(iii) Complete the parallelogram using the two vectors drawn in (i) and (ii) as two sides of the parallelogram.
(iv) The resultant force is represented in both magnitude and direction by the vector corresponding to the diagonal of the parallelogram drawn from the tail of the vectors in (i) and (ii)

FIG 4

1-3 Resultant of more than Two Coplanar Forces

Polygon of forces (using Bow’s notation).

The polygon of forces is used to solve any number of concurrent, coplanar forces acting on

a point P. They are concurrent because they act on a single point P. They are coplanar because

they lie in the same plane figure 5

● The forces acting on point P should be represented by vectors whose length and angles must be drawn to scale as accurately as possible. The spaces between the forces are designated using capital letters. The starting point is not important but the lettering must follow round consecutively, usually in a clockwise direction.

● The force diagram must be drawn strictly to scale, starting with a convenient vector. For example, start with the horizontal force F3. Since it lies between the spaces D and A this vector is labelled da. Remember that conventionally the spaces are given capital letters whereas the forces are given the corresponding lower case letters. Add the arrowhead to show the direction in which the force is acting.

● The next force (F4) lies between the spaces A and B. It is drawn to scale following on from the end of the first vector and labelled ab. Again add the arrowhead to indicate the direction in which the force is acting.

● Similarly, add the remaining forces so that they follow on in order labelling them accordingly.

● Finally, the resultant force (FR) can be determined by joining the points and in the force diagram. Note that the direction of the resultant force is against the general flow of the given forces in the force diagram. If required, the resultant force can now be transferred to the space diagram to show its magnitude and direction relative to the point P.

● Remember that the resultant force can replace all the given forces yet still have the same effect on the point P.

● If the equilibrant had been required then it would have the same magnitude as the resultant force but it would act in the opposite direction. It would follow the general flow of the given forces in the force diagram.

FIG 5

2 - Equilibrant forces

A force which cancels out the effect of another force or system of forces is called the equilibrant force (FE). An equilibrant force (figure 6):
● has the same magnitude as the resultant force,
● has the same line of action as the resultant force,
● acts in the opposite direction to the resultant force.

FIG 6

2-1 Three forces in equilibrium
● The vectors of three forces acting in the same plane on a body can be drawn on a flat sheet of paper.
● Forces that act in the same plane are said to be coplanar.
● If the three forces acting on a body are in equilibrium, any one force balances out the effects of the other two forces. The three forces may be represented by a triangle whose sides represent the vectors of those forces.
● If the lines of action of the three forces act through the same point, they are said to be
concurrent.
AS in Figure 7 shows three coplanar forces acting on a body. The point P is called the point of concurrency. If the forces are in equilibrium:
● Their effects will cancel out.
● The body on which they are acting will remain stationary.
● The vectors of the forces will form a closed triangle.
● The vectors follow each other in a closed loop.

FIG 7

3- Resultant of Coplanar Forces by Calculation
An alternative to the graphical methods of determining the resultant of two coplanar forces is by calculation. This can be achieved by trigonometry using the cosine rule and the sine rule, or by resolution of forces

FIG 8

The resultant is given by length ob. By the cosine rule

ob² = oa²  + ab² - 2  (oa) (ab)  cos (∟ oab)

                                      = 8² + 5² -  2 ×8×5  cos (100°)

4- Resolution of forces
The resolution of forces is the reverse operation to finding the resultant force(figure 9). That is, the
resolution of a force is the replacement of a given single force by two or more forces acting at the same point in specified directions.

FIG 9

To resolve the force into its horizontal and vertical component forces first draw the lines of action of these forces from the point P as shown in Fig. 9(b). Then complete the parallelogram of forces. The magnitude of the forces FV and FH can be obtained by scaling the drawing (vector diagram) or by the use of trigonometry