If a number of force are acting on a body at the same point then these forces are called concurrent and if these forces are unable to change state of the body i.e.,body remains in its state or rest or of uniform motion along a straight line,then the body is in the state of equilibrium and so are these forces.
Thus,the number of forces acting on a body may be in the state of equilibrium if these forces produce zero resultant force.
Let us consider three concurrent forces F1(vector),F2(vector) and F3(vector) acting simultaneously at the same point O.
Let force F1(vector)and F2(vector) be represented by two sides of parallelogram,complete the parallelogram.Find the resultant of these two forces F1(vector)and F2(vector),this will act along OC' i.e.,along the diagonal of the parallelogram.
→ → →
Therefore R = F1 + F2
It this resultant is equal and opposite to the force F3,then the resultant three of forces
F1(vector),F2(vector)and F3(vector) acting at point O will be zero.
→ → → →
i.e.,if OC = -OC' i.e., F3= -R
→ → → → → →
or if F3=-(F1+ F2) or F1+F2+F3=0.
Hence these three forces F1(vector),F2(vector)andF3(vector)are in equilibrium.Similarly if no.of forces F1(vector),F2(vector),F3 ........,Fn act on body at same point,then the body will in equilibrium state iff.
→ → → →
F1+F2+F3+.......+Fn = 0.
Lami's theorem:
It states that when three forces acting at a point are in equilibrium.each of the forces is proportional to the sine of the angle enclosed between the other two forces.
Proof : suppose P,Q and R are three forces in equilibrium acting at a point O.These forces can be represented by a triangle ABC,the side ABC of which is parallel to R,BC parallel to P and CA parallel to Q.
Therefore P/BC = Q/AC = R/AB
Or P/a=Q/b=R/c
Where,a,b and c are the triangle opposite to the angles A,B and C respectively.
In any triangle ABC,we have
a/sinA=b/sinB=c/since (sine formula)
Now (Angle)A =180°-a,
(Angle)B =180°-B and (Angle)C = 180°-Y
SinA = sin(180°-a) = sin a
SinB = sin(180°-b) = sin b
SinB = sin(180°-y) = sin y
Therefore a/sin a=b/sin b=c/sin y
From(i) and(ii),we get,
P/sin a = Q/sin b = R/sin y
Thus,each force is proportional to the angle between the other two forces.This proves Lami's theorem.
Thus,the number of forces acting on a body may be in the state of equilibrium if these forces produce zero resultant force.
Let us consider three concurrent forces F1(vector),F2(vector) and F3(vector) acting simultaneously at the same point O.
Let force F1(vector)and F2(vector) be represented by two sides of parallelogram,complete the parallelogram.Find the resultant of these two forces F1(vector)and F2(vector),this will act along OC' i.e.,along the diagonal of the parallelogram.
→ → →
Therefore R = F1 + F2
It this resultant is equal and opposite to the force F3,then the resultant three of forces
F1(vector),F2(vector)and F3(vector) acting at point O will be zero.
→ → → →
i.e.,if OC = -OC' i.e., F3= -R
→ → → → → →
or if F3=-(F1+ F2) or F1+F2+F3=0.
Hence these three forces F1(vector),F2(vector)andF3(vector)are in equilibrium.Similarly if no.of forces F1(vector),F2(vector),F3 ........,Fn act on body at same point,then the body will in equilibrium state iff.
→ → → →
F1+F2+F3+.......+Fn = 0.
Lami's theorem:
It states that when three forces acting at a point are in equilibrium.each of the forces is proportional to the sine of the angle enclosed between the other two forces.
Proof : suppose P,Q and R are three forces in equilibrium acting at a point O.These forces can be represented by a triangle ABC,the side ABC of which is parallel to R,BC parallel to P and CA parallel to Q.
Therefore P/BC = Q/AC = R/AB
Or P/a=Q/b=R/c
Where,a,b and c are the triangle opposite to the angles A,B and C respectively.
In any triangle ABC,we have
a/sinA=b/sinB=c/since (sine formula)
Now (Angle)A =180°-a,
(Angle)B =180°-B and (Angle)C = 180°-Y
SinA = sin(180°-a) = sin a
SinB = sin(180°-b) = sin b
SinB = sin(180°-y) = sin y
Therefore a/sin a=b/sin b=c/sin y
From(i) and(ii),we get,
P/sin a = Q/sin b = R/sin y
Thus,each force is proportional to the angle between the other two forces.This proves Lami's theorem.