Verify the Triangle Law of Forces and Lami’s Theorem

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TRIANGLE LAW OF FORCES AND LAMI’S THEOREM

FORCE

Definition of force: A force can be defined as an agent which has a tendency ” to generate or to kill ” a motion.

for example: 1.we push a table, here the force applied by us to generate motion.
2. we try to stop a bicycle by pulling it toward us, here we kill the motion of the bicycle by pulling it toward us with some force.

Let us go through some technical terms regarding force.

1. coplanar forces: co planar forces is defined as the force or a set of forces acting a body are in the same geometric plane.

2. Concurrent forcse: the forces acting or meeting in a single point of action is said to be concurrent force.

3. Non-Co-linear forces: Non-co-linear force is defined as the force which doesn’t act on the same line of action of that force.

Resolution of Forces: Resolution of forces is said to be resolving of forces in to its smaller units, or it also can be defined in simple words ” splitting of the forces without changing their influence on th body”. Resolution of forces is generally carried out on two mutually perpendicular componenet i.e. vertical component along Y-axis and Horizontal component along X-axis.

Analytical resolution of force into vertical and horizontal component.

let us take a force of magnitude ‘F’ acting at angle “30 ͦ ” to horizontal plane ‘V’.
Similarly, the same force has it effect on the vertical component plane( which is mutually perpendicular to the horizontal plane). and it makes an sngle of 60 ͦ with it ( as 90 ͦ -30 ͦ ).

now the value of the component of force along horizontal axis being ‘ F x Cos30’
similarly, its component to vertical axis is

X= ‘F x Cos(90-30)’
as its a complimentary angle so it can be written as,
Y= ‘F x Sin(30)’

here by applying Pythagoras theorem we can say, that the the resultant would be the hypotenuse w.r.t to vertical component and horizontal component.
let the resultant component would be ‘R’
so,
R= √(ΣX^2 +ΣY^2)

and the angle of action of resultant force is given by ‘ θ ‘

tanθ=(ΣY/ΣX).

TRIANGLE LAW OF FORCE.

According to this law if there are three forces acting over a body which is in equilibrium. then, the two forces are represented as a two side of a triangle in same order with scaling their magnitude to a suitable scale then the third side or the closing side of the triangle would be the resultant in opposite order.

Triangle law of forces

for example : let us take two forces of magnitude 50N and 100N acting on a same geometric plane at an angle of 30 ͦ and 60 ͦ respectively.

here, we can solve the following by triangle law of forces.

first scale the forces into a suitable scale,then
step 1-from the point ‘o’ draw the first force(OA) value converted into scale at an angle of 30 ͦ made by the force with horizontal axis.
step 2- taking the first force (‘OA’) as the horizontal axis now draw the second force(‘AB’) at an angle of 60 ͦ made with the OA as axis.
step 3- now the point B is joined with the point O. The value or distance is calculated and then converted into Newton by multiplying the scale taken for drawing.

here the side of the triangle OB is the resultant of the two forces applied on a body without altering their effect on the body.

LAMI’S THEOREM.

This theorem was given by a Great Mathematician “Bernard Lamy”, and the name was coined Lami’s theorem.

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.
( A/Sin α) = ( B/Sin β) = (C/ Sin γ)

also we can solve it by triangle law,

proof of lami’s theorem

by the above empirical formulae we can calculate the magnitude of force on the body.

Important questions:
Q- Define lami’s theorem?
Ans-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.

Q- Into how many component a force can be ressolve?

Ans- One can rsolve the force into two of its component i.e. vertical component and horizontal component without altering the effect of force on the body on which it is applied.

Q- Solve the following diagram by resolution of forces method.

Ans- the tension in the string due to the weight W is same at point c i.e. T3=T4 and here we can resolve the force into two component of it i.e. vertical and horizontal componenet as we know that upward force is equal to the force acting downward as the body is in equilibrium,similarly with the horizontal componenet and again solving the componenet of T1 along vertical and horizontal componenet and solve as stated earlier

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