Question

The maximum and minimum magnitudes of the resultant of two forces are 35 N and 5 N respectively. Find the magnitude of resultant force when act orthogonally to each other.

Answer 1

Hint: The individual vector values are found by equating the maximum and minimum magnitudes of the forces given. We will make use of the resultant vector formula to find the value of the magnitude of the resultant force.

Formula used:
\[R=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\]

Complete step-by-step answer:
The formula being used is as follows:
The resultant of the two vectors is given by the formula as follows.
\[R=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\]
Where A, B are the vectors and \[\theta \] is the cos of the angle between the vectors.
From given, we have that data,
The maximum and minimum magnitudes of the resultant of two forces are 35 N and 5 N respectively.
\[\begin{align}
  & \Rightarrow {{R}_{\max }}=35\,N \\
 & \Rightarrow {{R}_{\min }}=5\,N \\
\end{align}\]
Let A and B be the forces mentioned.
The maximum magnitude of the two forces is calculated as follows.
\[\begin{align}
  & {{R}_{\max }}=A+B \\
 & \Rightarrow A+B=35\,N \\
\end{align}\]…… (1)
The minimum magnitude of the two forces is calculated as follows.
\[\begin{align}
  & {{R}_{\min }}=A-B \\
 & \Rightarrow A-B=5\,N \\
\end{align}\]…… (2)
Solve the equations (1) and (2) to obtain the individual values of the magnitudes of the forces A and B. So, we have,
\[\begin{align}
  & 2A=40 \\
 & \Rightarrow A=20\,N \\
\end{align}\]
Substitute this value of the force in one of the two equations (1) and (2) to find the value of the vector B.
\[\begin{align}
  & 20\,-B=5 \\
 & \Rightarrow B=15\,N \\
\end{align}\]
The magnitude of resultant force when these forces A = 20 N and B = 15 N act orthogonally to each other is calculated using the formula as follows.
\[R=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\]
Substitute the values of the magnitudes of the individual forces obtained in the above equation. So, we get,
\[\begin{align}
  & R=\sqrt{{{20}^{2}}+{{15}^{2}}+2(20)(15)\cos 90{}^\circ } \\
 & \Rightarrow R=\sqrt{400+225+0} \\
\end{align}\]
Continue the further calculation.
\[\begin{align}
  & R=\sqrt{625} \\
 & R=25\,N \\
\end{align}\]
Therefore, the magnitude of resultant force when the two forces act orthogonally to each other is 25 N.

Note: In order to solve these types of questions, the formulae for calculating the maximum and the minimum magnitudes of the vectors should be known. The units of the parameters should be taken care of. As in the question statement, both the forces are given in the terms of Newton, so no need to convert.