COMPOSITION AND RESOLUTION OF FORCES

COMPOSITION AND RESOLUTION OF FORCES

DEFINITIONS

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RESULTANT FORCE:-

Resultant force is that single force, which produces the same effect as the effect, produced by the combination of several forces.

EQUILIBRIUM FORCE:-

It is the single force, which sets in the opposite direction to the resultant and keeps it in equilibrium.

COMPOSITION OF FORCES:-

It is the single force, which sets in the opposite direction to the resultant and keeps it in equilibrium.

RESOLUTION OF FORCES:-

The process of inverse of composition of forces is called resolution of forces. This is the process of splitting a force into two or more component forces.
A force is generally resolved along two mutually perpendicular directions.

ANALYTICAL METHOD OF FINDING OUT RESULTANT FORCE

       Methods of Resultant Force       LAW OF PARALLELOGRAM OF FORCES    

LAW OF PARALLELOGRAM OF FORCES:-

It states that if two concurrent forces are represented in magnitude, direction and sense by two adjacent sides of a parallelogram, the diagonal of the parallelogram, concurrent with the two forces, gives the resultant in magnitude, direction & sense.

PROOFExpression for the resultant of two forces acting at a point
Let P and Q be the two forces acting at a point O, at an angle Æ between them P and Q are represented both in magnitude and direction by OA and OB respectively.

Consider OA and OB as two adjacent sides, of parallelogram OACB is completed. The diagonal OC now gives the resultant R. the line OA is extended and a perpendicular CD is drawn from C on to it. Let a be the angle between P and R.

Magnitude of resultant OC
From the  triangle OCD
OC² = OD² + CD²
       = (OA+AD) ² + CD²
= OA² + AD² +2.OA.AD + CD²
    OA² + 2.OA.AC.cos Æ. + AC²
R² = P² + Q² +2PQcosÆ.
R=Ö P² + Q² +2PQcosÆ.
Direction of the resultant
Here R gives the magnitude
From the triangle OCD
Tan a = CD / OD
          =  CD / (OA+AD)
           =AC sin Æ / (P+ Q cos Æ)\ a = Tan^-1 (Q sinÆ) /(P+Q cos Æ)
Thus, a gives the direction of the resultant.

LAW OF TRIANGLE OF FORCES:-

It states that if two concurrent forces are represented in magnitude, direction and sense by two sides of a triangle taken in order, their resultant is represent by the third side of the triangle in magnitude and direction but in opposite sense.

LAW OF POLYGON OF FORCES:-

It states if a number of concurrent forces are represented in magnitude, direction and sense by the sides of an open polygon, taken in order their resultant is represented in magnitude and direction by the closing side of the polygon but in the opposite sense.

PROBLEMS:-1.Find the resultant of the vector represented by 9N, 10N acting in the direction and making angle of 60⁰.
Resultant
R²  = P² + Q² + 2PQ Cos. q
      = 9²  + 10² + 2 x 9 x 10 x Cos 60⁰
= 81 + 100 + 12 X 9 X 10 x Cos 60⁰       R²  = 271R  = 16.46 N
  2.A fitter holds a chisel at an angle of 45⁰ to the horizontal and strikes it a blow of 150 N. Find the force tending to drive the chisel vertically into the material and the force tending to drive the chisel horizontally.
  Solution:   Force tending to drive chisel horizontally into the material = Horizontal component\ H = 150 x cos 45⁰        = 150 x 1 / Ö2 kg wt
Similarly, force tending to drive the chisel vertically into the material = vertical component
V = 150 x sin45⁰
   = 150 x 1 / Ö2 kg wt

PROBLEMS BASED ON METHOD OF RESOLUTION1.An Aeroplane flies 20.0 KM in a direction 60⁰ north of east, then 30.0 KM straight east, than 10.0Km straight north. How far and in what direction is the plane from the starting point?
 
Solution:The situation is shown in fig. We have chosen the X-axis as the east and the y-axis as the first displacement, B the second .C the third. And R the vector sum or resultant displacement. The components of A are:
Ax = 20 cos 60⁰ = 10.0 Km
Ay = 20 sin 60⁰  =  17.3 Km
The components of all the displacement and the calculations can be arranged systematically, as in table.