Question

A body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is:
(a) 2, if their lines of action pass through the center of mass of the body.
(b) 3, if their lines of action are not parallel.
(c) 3, if their lines of action are parallel.
(d) 4, if their lines of actions are parallel and all the forces have the same magnitude.

Answer 1

As in the question it is given that a body is in equilibrium under the influence of a number of forces with each force having a different line of action so, to find the number of forces required for an object to be in equilibrium we will use the concept of the equilibrium under number of forces as zero, one, two and so on. This concept will also include the magnitude as well as direction.

Formula used: $\left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=\sqrt{{{s}^{2}}+{{s}^{2}}+2{{s}^{2}}\cos \theta }$ where, $\overrightarrow{P}$ and $\overrightarrow{Q}$ are two forces such that their magnitudes are equal to $\overrightarrow{s}$ like so, $\left| \left. \overrightarrow{P} \right| \right.=\left| \left. \overrightarrow{Q} \right| \right.=\overrightarrow{s}$.

Complete step-by-step answer:
First of all, we will understand the concept of the number of forces under equilibrium.
The equilibrium under no force: Here, the equilibrium exists as there is no force exerted on the object.
The equilibrium under 1 force: Under one force no object can attain its equilibrium state as the magnitude of any object with only one force applied on it will exert a force of $F=\sqrt{{{x}^{2}}+{{y}^{2}}+2xy\cos \theta }$. Here, the x and y coordinates are the coordinates that the object makes.
The equilibrium under 2 forces: Equilibrium under two forces can be attained by an object if one force is acting on the object with an equal and opposite force acting on the same object. For example if we take two forces as $\overrightarrow{P}$ and $\overrightarrow{Q}$ such that their magnitudes are equal to $\overrightarrow{s}$. This means that $\left| \left. \overrightarrow{P} \right| \right.=\left| \left. \overrightarrow{Q} \right| \right.=\overrightarrow{s}$. Then, the addition of these two forces will be equal to the magnitude $\left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=\sqrt{{{s}^{2}}+{{s}^{2}}+2{{s}^{2}}\cos \theta }$. Since, the forces are equal and opposite so, they will form an angle of ${{180}^{\text{o}}}$. Therefore, we have
$\left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=\sqrt{{{s}^{2}}+{{s}^{2}}+2{{s}^{2}}\cos \left( {{180}^{\text{o}}} \right)}$. As $\cos \left( {{180}^{\text{o}}} \right)=-1$ so,
$\begin{aligned} & \left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=\sqrt{{{s}^{2}}+{{s}^{2}}+2{{s}^{2}}\left( -1 \right)} \\\ & \Rightarrow \left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=\sqrt{2{{s}^{2}}-2{{s}^{2}}} \\\ & \Rightarrow \left| \left. \overrightarrow{P}+\overrightarrow{Q} \right| \right.=0 \\\ \end{aligned}$
This results in the two forces at equilibrium having the same magnitude.
The equilibrium under 3 forces: Similar by the case of the equilibrium under 2 forces, if three forces are acting on an object, then for the object to be in equilibrium must have a force which is equal to the addition of the two forces resulting into the magnitude of the addition of two forces to be equal to the addition of the third force.
The equilibrium under 4 forces: If the four forces having an equal and opposite forces as well as having equal magnitudes resulting into zero will imply an object to be in equilibrium.
Now, according to the four options the fourth option is the correct one as this clearly states about the magnitude along with the direction of the force. Moreover, force is the vector quantity.
Hence, the correct option is (d) stating that a body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is 4, if their lines of actions are parallel and all the forces have the same magnitude.

Note: The things to be on your finger tips for further questions like this are,
(1) The equilibrium is attained when the forces acting on the object are equal, opposite and have the same magnitude.
(2) Force is a vector quantity; direction and magnitude of the forces are important to be noticed.
(3) The minimum number of forces: 2, if the lines of action pass through the center of mass of the body along with the equal magnitude.
(4) The number of forces: 3, if their lines of action are parallel with considering the magnitudes simultaneously.