Name: Lami's Theorem in Equilibrium
Uploaded: 2019-07-19
Duration: 7 min 28 s
Description: In this interesting video, we will learn about Lami's theorem in equilibrium. Watch the solved sample problems to quickly understand the theorem.

Question

State triangle law of vectors, and derive an expression for the resultant of the vectors

\vec{A}

and

\vec{B}

.

Accepted Answer

Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector

Consider two vectors $\vec{A}$ and $\vec{B}$ that are represented in the order of magnitude and direction by the sides OA and AB, respectively of the triangle OAB. Let $\vec{R}$ be the resultant of vectors $\vec{A}$ and $\vec{B}$

Let

$\vec{A} = \vec{P} \vec{B} = \vec{Q}$

From $∆ O C B$ ,

$O B^{2} = O C^{2} + B C^{2} O B^{2} = {(O A + A C)}^{2} + B C^{2} . . . . . . . . . . . . . (i)$

In $∆ A C B$ with $θ$ as the angle between P and Q

$\cos θ = \frac{A C}{A B} \Rightarrow A C = A B \cos θ = Q \cos θ \sin θ = \frac{B C}{A B} \Rightarrow B C = A B \sin θ = Q \sin θ$

Substituting the values of $A C$ and $B C$ in equation (i)

$R^{2} = {(P + Q \cos θ)}^{2} + {(Q \sin θ)}^{2} \Rightarrow R^{2} = P^{2} + 2 P Q \cos θ + Q^{2} \cos^{2} θ + Q^{2} \sin^{2} θ \Rightarrow R^{2} = P^{2} + 2 P Q \cos θ + Q^{2} \Rightarrow R = \sqrt{P^{2} + 2 P Q \cos θ + Q^{2}}$

This is the magnitude of the resultant vector.

To determine the direction of the resultant vector, let $ϕ$ be the angle between resultant vector $\vec{R}$ and $\vec{P}$ .

From $∆ O B C$ ,

$\tan ϕ = \frac{B C}{O C} = \frac{B C}{O A + A C} \tan ϕ = \frac{Q \sin θ}{P + Q \cos θ} ∴ ϕ = \tan^{- 1} (\frac{Q \sin θ}{P + Q \cos θ})$

This is the required expression for direction of the resultant.

State triangle law of vectors, and derive an expression for the resultant of the vectors $\vec{A}$ and $\vec{B}$ .

Important Questions on Scalars and Vectors

Learn and Prepare for any exam you want !

State triangle law of vectors, and derive an expression for the resultant of the vectors A→ and B→.

Important Questions on Scalars and Vectors

Learn and Prepare for any exam you want !

State triangle law of vectors, and derive an expression for the resultant of the vectors $\vec{A}$ and $\vec{B}$ .