MTH6P2B: COPLANAR FORCES

Coplanar forces means the forces in a plane. When several forces act on a body, then they are called a force system or a system of forces. In a system in which all the forces lie in the same plane, it is known as coplanar force system.

Hence this article deals with a system of forces which are acting in the same plane and the forces are either having a common line of action or intersecting at a common point. If the forces are having common line of action, then they are known as collinear whereas if the forces intersect at a common point, then they are known as concurrent.

A force system may be coplanar or non-coplanar. If in a system all the forces lie in the same plane then the force system is known as coplanar. But if in a system all the forces lie in different planes, then the force system is known as non-coplanar. Hence a force system is classified as shown in Fig. 1.31.

If a system of coplanar forces is acting on a body, its total effect is usually expressed in terms of its resultant. Force being a vector quantity the resultant of the system of forces can bc found out by using vector algebra, e.g. if the resultant of two forces is to be found out then the law of parallelogram of forces is used.

Law of Parallelogram of Forces
If the two coplanar forces meet at a point, their resultant may be found by the law of parallelogram of forces, which states that, “If two forces acting at a point are such that they can be represented in magnitude and direction by the two adjacent sides of parallelogram, the diagonal of the parallelogram passing through their point of intersection gives the resultant in magnitude and direction”.

Consider two forces P and Q acting at point 0 in the body as shown in Figure 1.6 (a). Their combined effect can be found out by constructing a parallelogram using vector P and vector Q as two adjacent sides of the parallelogram as shown in Figure 1.6 (b). The diagonal passing through 0 represents their resultant in magnitude and direction. You can.

prove by :fie geometry of the figure that the magnitude R of the resultant and the angle it makes with P are given by

The above two forces can also be combined by using the law of triangle of forces which states that if the second force is drawn from the end of the first force then the line joining the starting point of first force to the end of the second force represents their resultant.

From the triangle of forces, by using trigonometric relations, you can find that

Law of Polygon of Forces
If more than two forces are acting on a body, then their resultant can be found by repeated applications of the parallelogram law or the triangle law. You may start with any tw0 forces and find their resultant first and then add vectorially to this resultant the remaining forces taking one at a time.

In the final form a polygon would be completed. In other words, if more than two coplanar forces meet at a point, their teedltMt my be found by the law of polygon of forces, which states that, “If a number of fmwe acting at
a point are such that they can be represented in magnitude and direction by the sides of 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”.

Example 1.3
Consider five forces each of 80 N acting at 0 in a body, Draw forces of polygon and show the resultant of dl the forces.

Solution

Let us construct a polygon such that the forces  represent the sides of a polygon taken in order, each force being drawn from the end of earlier force then their resultant is represented by the line joining the starting point of the first force to the end of the last force

SAQ 2
Determine the resultant in nlagnitucle and direction of two forces shown in Figures 1.8 (a) and (b) using the parallelogram law and the triangle law.

SAQ 3
Four forces are acting at 0 as shown in the Figure 1.9. Find the resultant in magnitude and direction by using polygon law.

Resolution-and Composition
In many engineering problems, it is desirable to resolve a force into rectangular components. ?his process of splitting the force into components is called the resolution of a force whereas the process of finding the resultant of any number of forces is called the composition of forces. ‘Ihe resolution of forces helps in determining the resultant of a number of forces acting on a body as it reduces vectorial addition to algebraic addition.
A force making an angle with respect to x axis as shown in Figure 1.10 can be
resolved into two components acting along x and y axes respectively. If and

are the unit vectors acting along x and y axes respectively then the force ji? can be expressed as

where are the magnitudes of the components along x and y axes. Refer Figure ,I. 10 are determined as

Note : is measured in anticlockwise direction with respect to positive x axis.
The magnitude of the force can also be expressed as

Example 1.4
A force of 120 N is exerted on a hook in the ceiling as shown in Figure 1.1 1. Determine the horizontal and vertical components of the force.

As is to be measured in anticlockwise direction from positive x axis, then

The vector components of force are

Example 1.5
A force of 80 N is acting on a bolt as shown in Figure 1.12. Find the horizontal and vertical components of the force.

Solution
By principle of transmissibility of a force, the force can be considered acting at any point on the line of action of the force.

with respect to positive x axis measured in anticlockwise direction

NON-COPLANAR FORCES

In engineering practice, we come across situations where the fo&s acting on a body may not be in one plane, e.g. a crate supported by three cables in space, a precast concrete wall section temporarily held by four cables or a tower guy wire anchored by means of bolts.

To solve problems of statics in space of three dimensions, a lot of visualisation is needed; as we have to draw three dimensional sketches on the familiar two dimensional plane of principles developed in two dimensions to three dimensional space.

sheet of paper. Vector notation will be useful in solving such problems by extending the 9,
Rectangular Components
Consider a force F acting at 0. Assume a system of rectangular coordinates ‘n, y and z with ‘0’ as the origin. To determine the direction of the force, let us construct a parallelepiped (say ‘box’) as shown in Figure 1.13 (a).

are the angles made by F with respect to x, y and z axes respectively. then we get :

The three angles define the direction of the force F and and are called the direction cosines of the forces which are also represented by I, m and n respectively.

If  are the unit vectors acting along x, y and z axes as shown in Figure 1.13 (b), then force F can be expressed in vector form as under :

and the magnitude of F is given by

Example 1.6
A force of 400 N forms angles of  respectively with the x, y and z axes. Express the force in vector form.

Solution

Example 1.7
Determine the magnitude and direction of the force

Solution
The magnitude of the force F is given by

Thus, the force of 815.41 N is making an angle of  with x axis,  with y axis and with z axis.

RESULTANT OF CONCURRENT FORCES

The resultant of a concurrent force system can be defined as the simplest single force which can replace the original system without changing its external effect on a rigid body.
For the 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 types of force systems as classified in Section 1.2 along with their possible resultants are given in Table 1.1.

We will study the determination of resultant of non concurrent force systems in Section 1.7, after knowing about moment, couple and their properties.

Resultant of Coplanar Concurrent Forces
The technique of resolution of a force can be used to determine the resultant of coplanar concurrent forces. If ‘n’ concurrent forces are acting at a point in a body then each force can be resolved into two mutually perpendicular directions. Thus, we get ‘2n’ components. Each set of 2n components acts in one direction only. Therefore, we can algebraically add all these components to get the components of the resultant.

Finally, combining these components vectorially, we get the resultant

where, is the angle of inclination of the resultant  with respect to positive x axis

Example 1.8
Four forces act on a body as shown in Figure 1.14. Determine the resultant of the system of forces.

Resultant of Non-coplanar Concurrent Forces

SAQ 4
(1) A hoist trolley is subjected to three forces as shown in Figure 1.15. Determine the magnitude and direction of the resultant.
(2) Solve SAQ 3 by the method of resolution of forces.

MOMENT OF A FORCE

If a force F is acting on a body resting at 0 and the line of action of fado es not pass through G, the centre of gravity of the My, it will not give the body a straight line triotiom, called the Qanslatory motion. but will try to rotate the body about 0 as shown in Figure: 1 15.

The measure of this property of a force by virtue of which it tends to rotate on the OR which it acts is called the moment of a force, The rotation of the body may be either  about a point or a line.

Referring the Figure 1.17, if F is the force (in N) acting on the body dong AB and x is the
perpendicular distance (in m) of 0 from AB, then

Moment of the force Fa bout O = M = Fxx = FxOC
Here, point 0 is known as moment centre or fulcrum and distance X is termed as moment arm.

If the moment of the force about a point is zero, it means either the force itself is zero 01′ the perpendicular distance between the line of action of the force and the point about which moment is to be calculated is zero i.e the force passes through that point.
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

Moment of Coplanar Forces

Now, the magnitude and direction of resultant R can be found out very easily by resolving all the forces horizontally and vertically as discussed in Section 1.5.1, Let the resultant R makes an angle 0 with positiva axis As shown in Figure 1.18 (b). Now by computation of moment of forces, the position of resultant force R can be ascertained.

Moment of a Force about a Point and an Axis

The moment of a force can be determined with respect to (about) a point and also with respect to a line or axis.
The moment of a force F with respect to a point A is defined as a vector with a magnitude equal to the product of the perpendicular distance from A to F and the magnitude of the force and with a direction perpendicular to the plane containing A and F  The sense of the moment vector is given by the direction a right-hand screw would
advance if turned about A in the direction indicated by F as shown in Figure 1.19.

The moment of a force about a line or axis perpendicular to a plane containing the force is defined as a vector with a magnitude equal to the product of the magnitude of the force and the perpendicular distance from the line to the force and with a direction along the line. Thus, it is the same as the moment of the force about the point of intersection of the plane and the moment axis.

Since the moment of a force about an axis is a measure of its tendency to turn or rotate a body about the axis, the force parallel to an axis has no moment with respect to the axis, because it has no tendency to rotate the body about the axis.

The moment of a force about various points and axes is illustrated in Figure 1.20. The moment of the horizontal force F about point A has a magnitude of in the direction.

shown by (Figure 1.20 (a)). Similarly the moment about point B, has a magnitude of  and is perpendicular to the plane determined by B and the force F. The moment of force F about the line AB is the same as (as shown in Figure 1.20 (a)) or (as shown in Figure 1.20 @)).,The moment of force F about line BC can be obtained by resolving force F into components is parallel to line BC, it has no moment about BC. The resultant is in a plane perpendicular to BC and its moment is in the direction shown. Similarly, the moment about line BD is as indicated. Here, you can note that MAE, MBc and MBD are the orthogonal components of ME.

Example 1.9
The side of a square ABCD is 1.60 m long. Four forces equal to 6,5,4 and 8 N act along the line CB, BA, DA and DB respectively. Find the moment of these forces about 0, the point of intersection of the diagonals of the square.

SAQ 5
The side of a regular hexagon ABCDEF is 0.6 m. Forces l,2,3,4,5 and 6 N are acting along the sides AB, CB, DC, DE, EF and FA respectively. Find the algebraic sum of the moments about A

Couples ax! their Properties
A couple is a force system consisting of two equal, coplanar, parallel forces acting in opposite direction. Since a couple constitutes two equal and parallel forces, their resultant is zero and hence a couple has no tendency to produce translatory motion but produces rotation in the body on which it acts.

Figure 1.23 shows two equal and opposite forces, each equal to P and acting at A and B along parallel lines, thus constituting a couple. The perpendicular distance AB is called the arm of the couple and is denoted by p.
Moment of a Couple.

The moment of a couple about any point in the plane containing the forces is constant ansa is measured by the product of any one of the forces and the perpendicular distance between the lines of action of the forces, i.e., M = P x p .
Properties of Couples
The properties which distinguish one couple from every other couple are called its characteristics. A couple whether positive or negative, has the following properties/characteristics.

The algebraic sum of the forces constituting a couple is zero.
The algebraic sum of the moment, of the forces forming a couple is the same , about any point in their plane.
The couple can be balanced only by another couple of the same moment but of the opposite sense.
The net effect of a number of coplanar couple is equivalent to the algebraic sum of the effect, of each of the couples.
A couple is frequently indicated by a clockwise or counterclockwise arrow when coplanar force systems are involved instead of showing two separate forces.
Replacement of a Force by a Force and Couple

Replacement of a Couple by two Forces
Consider a couple of moment M, where the axis of the couple is through 0 perpendicular to the plane of paper as shown -in Figure 1.25.

This couple is equivalent to any two parallel forces of magnitude F acting at a distance d apart such that F.d = M and the directions of the forces so chosen as to give the correct direction of M.

RESULTANT OF NON-CONCURRENT FORCES

As stated earlier, the resultant of a system of forces is the simplest force system which can replace the original forces without altering their external effect on a rigid body. The equilibrium of a body is the condition wherein the resultant of all the forces, is zero. The properties of force, moment and couple discussed in the preceding sections will now be used to determine the resultants of non concurrent force systems.

Resultant of Coplanar Non-concurrent Forces
The resultant of a system of coplanar non concurrent forces may be obtained by adding two forces at a time and then combining their sums. The three forces Fl, F2 and F3 shown in Figure 1.26 may be combined by first adding any two forces such as F2 and F3 . They may be moved along their lines of action to their point of concurrency A by the principle of transmissibility.

Their sum is formed by the law of parallelogram of forces. The force may then be combined with by the parallelogram law at their point of concurrency B to obtain the resultant R of the three given forces. Here, the order of combination of the forces is immaterial as may be verified by combining them in a different sequence. Now, the force R may be applied at any point on its established line of action.

Algebraically, the same result may be obtained by forming the rectangular components of the forces in any two convenient perpendicular directions. In Figure 1.26 (b), the x and y components of R are seen to be the algebraic sums of the respective components of the three forms. Thus, in general, the rectangular components of the resultant R of a coplanar system of forces may be expressed as

For this system of forces where the clockwise direction has been taken as positive, the distance d is computed from this relation, and R , whose magnitude and direction have already been determined earlier, may now be completely located. In general, then, the moment arm d of the resultant R is given by

For a system of parallel forces, the magnitude of the resultant is the algebraic sum of the several forces, and the position of its line of action may be obtained from the principle of moments.
Now, consider a force system such as shown in Figure 1.28, where the polygon of forces closes and consequently there is no resultant force R . Direct combination by the law of parallelogram shows that for the case illustrated, the resultant is a couple of magnitude F3d. The value of the couple is equal to the moment sum about any pint. Thus, it is seen that the resultant of a non-concurrent coplanar system of forces may be either a force or a couple.

Resultant of Non-coplanar Non-concurrent Forces
As mentioned earlier, there are many engineering problems which require analysis of non coplanar non concurrent system of forces which is three dimensional in nature. Such
analysis calls for representation of the system by a pictorial drawing or by means of two
or more orthographic projections.
The resultant of a non-parallel, non-coplanar, non-concurrent force system can be a single force or a couple, but in general it is a force and a couple. When all the forces of the system are parallel, the resultant will be a single force (parallel to the given forces) or a couple in the plane of the system or in parallel plane or the

the direction of the resultant couple may be specified by its direction cosines, which are

 

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