Prove the parallelogram law of forces - mechanics, Mechanical Engineering

Prove the parallelogram law of forces:

Assume that the two forces P and Q act at a point 'O' as shown in figure given below. The force P can be represented in magnitude and direction by vector OA, While force Q is represented in magnitude and direction by vector OB, Angle between the two force is 'a'. The resultant can be denoted by vector OC as shown in the figure given below. Drop perpendicular from C on OA.

Let,

P,Q = Forces whose resultant is needed to be found out.

θ= Angle that the resultant forces makes with one of the forces

α= Angle between the forces P and Q

1071_Prove the parallelogram law of forces.png

It is the magnitude of resultant 'R'

 

2383_Prove the parallelogram law of forces1.png

Direction ( θ ):

865_Prove the parallelogram law of forces2.png

 

Conditions

(i) Resultant R is max when two forces collinear and are in the same direction.

 2380_Prove the parallelogram law of forces3.png

(ii) Resultant R is min when two forces collinear but acting in the opposite direction.

That is  α = 1800    =>   Rmin = P- Q

(iii) If a = 900, that is when the forces act at right angle, then

R =   P2  + Q2

(iv) If the two forces are equal that is, when P = Q  =>R = 2P.cos(θ/2)

 

Posted Date: 10/16/2012 1:27:54 AM | Location : United States







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