Question

What is the minimum number of forces acting on an object in a plane that can produce a zero resultant force?

Answer 1

Hint: We have to find the least number of forces which upon their resultant give a zero value. It corresponds to a vector result, not a scalar quantity. The vector quantity is the one which is associated with the direction of vectors and thus two same and opposite forces cancel with each other.

Complete step by step answer:
Force is defined as an influence which when occurs causes a body to change its motion. A force causes a body with mass to have a change in its velocity. A force has both magnitude and direction, which makes it a vector quantity. In other words, force can be also considered as a push or pull.

According to Newton’s Second Law, the net force acting on a body is the time rate of change of its momentum. So, we can conclude that mass is constant, thus force is directly proportional to velocity.Equal in magnitude and opposite in direction vectors cancel each other, Thus, their resultant becomes zero.Force is also a vector quantity.A single force vector will have both magnitude and direction and hence it’s resultant will be itself which is not zero.

So, two force vectors of equal magnitude and opposite direction can make their resultant as zero.Let us consider each of the force vectors as $\overrightarrow{F}$ and opposite in direction, so the angle between them is ${180}^{\circ }$. Let the resultant of two vectors are $R$.Therefore, we get,
$$\begin{gathered} R=\sqrt{{\left(F\right)}^2+{\left(F\right)}^2+2.F.F\cos {180}^{\circ }} \\ \Rightarrow R=\sqrt{2F^2-2F^2} \\ \therefore R=0 \end{gathered}$$

Hence, the minimum number of forces so that their resultant becomes $0$ is $2$.

Note: It must be noted that if there are as many as forces if all are equal and opposite to others then their resultant will be zero. If only one vector is present, then it will follow a direction and hence have a magnitude of its own, so its resultant can never be zero.