To verify the law of
parallelogram of forces & the
Lami theorm
Answers
Answer:
The law of parallelogram of forces states that if two vectors acting on a particle at the same time be represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point .
Parallelogram law of forces
law of parallelogram of forces
Okay, got it what it wants to say. But, the main question here is how does it happen! Right?
So let us assume that the two vectors A and B, inclined at angle θ, be acting on a particle at the same time. Let they be represented in magnitude and direction by two adjacent sides OP and OS of parallelogram OPQS, drawn from a point O.
According to parallelogram law of vectors , their resultant vector will be represented by the diagonal of the parallelogram .
Magnitude and Direction of Resultant:
Parallelogram law of forces
law of parallelogram of forces
Draw a perpendicular QN to OP produced.
And let us assume that OP=A, OS= PQ= B, OQ=R and angle SOP= angle QPN = θ.
Now considering this if we proceed further , in the case of triangle law of vector addition , the magnitude and direction of resultant vector will be given by
R= sqrt of A^2 + B^2 + 2 AB cosθ
tan B = B sinθ/ A+B cosθ
Special cases :-
(1) When two vectors are acting in the same direction , then θ= 0 , cosθ=1 and sinθ= 0
R= sqrt of A^2 + B^2 + 2 AB
=sqrt of (A+B)^2 = A + B
tan Beta = B X 0/ A+B = 0
Beta = 0
Thus for two vectors acting in the same direction the magnitude of the resultant vector is equal to the sum of the magnitudes of two vectors and act along the direction of A and B.
(2) When two vectors are acting in opposite directions , then θ= 180 , cos θ= -1 and sinθ= 0
R= sqrt of A^2+ B^2+ 2 AB (-1)
= sqrt of (A-B) or (B-A)
tan beta = B X 0/ A+ B (-1)= 0
Beta = 0 or 180.
Thus for two vectors acting in opposite directions, the magnitude of the resultant vector is equal to the difference of the magnitudes of the two vectors and acts in the direction of bigger vector .
(3) When two vectors act at right angle to each other θ = 90 , sinθ = 1 and cosθ = 0
R= sqrt of A^2+B^2 + 2 AB (0)
= sqrt of A^2+B^2
tan beta = B(1)/A+B(0)= B/A
or, Beta = tan^-1 B/A
IMPORTANT NOTE :
1. It is to be noted that the magnitude of the resultant of two vectors is maximum , when the vectors act in the same direction and is minimum when they act in opposite directions.
2. It should be noted that while finding the resultant vector of two vectors by the parallelogram law of vector addition , the two vector A and B should be either act towards the point or away from the point .
Answer:
According to this theorem, when three coplanar, concurrent and non-co-linear forces act on a body which is in equilibrium then the magnitude of each force is proportional to the sine of angle between other two forces. This theorem can be proved by the sine law.