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Question: Two forces A and B act on an object in the opposite direction. A is bigger than B. The net force on ...

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+BA+B acting in the direction of A
B. A−BA-B acting in the direction of A
C. A+BA+B acting in the direction of B
D. A−BA-B acting in the direction of A
E. None of these.

Explanation

Solution

First using the formulas for vector addition calculate the magnitude of the resultant force. As the angle between the two forces is 180∘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 ϕ\phi and the angle between the forces A and B is θ\theta then the resultant of A and B has magnitude
∣A→+B→∣=A2+B2+2ABcos⁡θ\left| \overrightarrow{A}+\overrightarrow{B} \right|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }
tan⁡ϕ=Asin⁡θA+Bcos⁡θ\tan \phi =\dfrac{A\sin \theta }{A+B\cos \theta }

Complete step by step answer:
The magnitude of A and B is given by
∣A→+B→∣=A2+B2+2ABcos⁡θ =A2+B2+2ABcos⁡180 =A2+B2−2AB =(A−B)2 =A−B \begin{aligned} & \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{aligned}
Consider the angle between the resultant force and the bigger force A is ϕ\phi. As A and B are in opposite directions then the angle between A and B is θ=180∘\theta =180{}^\circ .
Then
tan⁡ϕ=Asin⁡θA+Bcos⁡θ=Asin⁡180A+Bcos⁡180=0 ⇒ϕ=0∘ \begin{aligned} & \tan \phi =\dfrac{A\sin \theta }{A+B\cos \theta }=\dfrac{A\sin 180}{A+B\cos 180}=0 \\\ & \Rightarrow \phi =0{}^\circ \\\ \end{aligned}
I.e. The resultant force makes an angle 0∘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−BA-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.