Question

Three forces acting on a particle are in equilibrium. The angle between the first and second is 90degree and that between the second and third is 120degree . Find the ratio of the force

Answer 1

So the angle between the first and the third forces will be,

$\displaystyle\longrightarrow\sf{360^{\circ}-(90^{\circ}+120^{\circ})=150^{\circ}}$

By Lami's Theorem, or the law of sines,

$\displaystyle\longrightarrow\sf{\dfrac{F_1}{\sin120^{\circ}}=\dfrac{F_2}{\sin150^{\circ}}=\dfrac{F_3}{\sin90^{\circ}}}$

Lami's theorem states that if three constant forces make a point in equilibrium, then the ratio of any one force among them to the sine of the angle between the other two forces is the same and is constant.

Therefore,

$\displaystyle\longrightarrow\sf{F_1:F_2:F_3=\sin120^{\circ}:\sin150^{\circ}:\sin90^{\circ}}$

$\displaystyle\longrightarrow\sf{F_1:F_2:F_3=\dfrac{\sqrt3}{2}:\dfrac{1}{2}:1}$

$\displaystyle\longrightarrow\sf{\underline{\underline{F_1:F_2:F_3=\sqrt3:1:2}}}$

Answer 2

Given:

Three forces and angle between them as $90^{o}$ and $120^{o}$.

To find:

Ratio between the forces.

Solution:

We have been given that 3 forces act on a particle and are in equilibrium. Let, the forces be $F_{1}$, $F_{2}$ and $F_{3}$.

It is also given that the angle between $F_{1}$ and $F_{2}$ is $90^{o}$ and between $F_{2}$ and $F_{3}$ is $120^{o}$.

For all the forces to be in equilibrium, the angle between $F_{1}$ and $F_{3}$ should be $150^{o}$ as per the diagram. (attached below)

Now,

The force $F_{3}$ could be resolved into horizontal and vertical components, that are $F_{3}cos60$ and $F_{3}sin60$ respectively.

Since it is given that all the forces are in equilibrium, the sum of all the forces acting on the particle will be 0.

$F_{net}=0$

From the diagram, we can see that

$F_{3}sin60=F_{1}$       $;$ $or$

$\frac{F_{1} }{F_{3} }=\frac{\sqrt{3} }{2}$             $(i)$

Also, we can see that

$F_{3}cos60=F_{2}$       $;$ $or$

$\frac{F_{2} }{F_{3} } =\frac{1}{2}$

Therefore, we see that

$F_{1}: F_{2} :F_{3}=\sqrt{3}:1:2$

Final answer:

Hence, the ratio of all the forces acting on the particle is $\sqrt{3}$ : 1 : 2.