Question

Two forces A and B act on an object in the opposite direction. A is bigger than B. The net force on the object is:
A. \[A+B\] acting in the direction of A
B. \[A-B\] acting in the direction of A
C. \[A+B\] acting in the direction of B
D. \[A-B\] acting in the direction of A
E. None of these.

Answer 1

Hint: First using the formulas for vector addition calculate the magnitude of the resultant force. As the angle between the two forces is $180{}^\circ $, so the direction of resultant force can be calculated from the vector law’s.

Formula used: If the angle between the resultant force and force A is $\phi$ and the angle between the forces A and B is $\theta$ then the resultant of A and B has magnitude
$\left| \overrightarrow{A}+\overrightarrow{B} \right|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }$
$\tan \phi =\dfrac{A\sin \theta }{A+B\cos \theta }$

Complete step by step answer:
The magnitude of A and B is given by
$\begin{align}
  & \left| \overrightarrow{A}+\overrightarrow{B} \right|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta } \\
 & =\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos 180} \\
 & =\sqrt{{{A}^{2}}+{{B}^{2}}-2AB} \\
 & =\sqrt{{{\left( A-B \right)}^{2}}} \\
 & =A-B \\
\end{align}$
 Consider the angle between the resultant force and the bigger force A is $\phi$. As A and B are in opposite directions then the angle between A and B is $\theta =180{}^\circ $ .
Then
$\begin{align}
  & \tan \phi =\dfrac{A\sin \theta }{A+B\cos \theta }=\dfrac{A\sin 180}{A+B\cos 180}=0 \\
 & \Rightarrow \phi =0{}^\circ \\
\end{align}$
I.e. The resultant force makes an angle $0{}^\circ $ with the Bigger force A. So the resultant force is in the same direction as of A.
So the resultant will have magnitude $A-B$ and will be in the same direction as of A.

So, the correct answer is “Option B”.

Note: The quantity which has both magnitude and direction is called a vector quantity. But even if current has both magnitude and direction, it is a scalar quantity. In vector addition and subtraction the vector laws should be followed. And for direction of the resultant vector the triangle law and parallelogram law should be used.