Question

Show diagramatically how force of 7N and 9N combine to give a resultant force of a.16N b.2N

Answer 1

a)both are in the same direction

so the forces should be added

b)both are in opposite directions

so the forces should be subtracted

Answer 2

Answer: When the forces are applied in the same direction the resultant of the two forces are 16N.

Answer: When the forces are applied in the opposite direction the resultant of the two forces are 2N.

Explanation:

Case 1-

When the two forces are applied along the same direction i.e. the angle between the two vector is zero then they are added algebriclly to give a resultant maximum as

By parallelogram law of vector addition we get the resultant-

$\overrightarrow F_{1} +\overrightarrow F_{2} =9 N+7N=16$

$R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} +2\overrightarrow F_{1}\overrightarrow F_{2}Cos\theta\\\\R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} +2\overrightarrow F_{1}\overrightarrow F_{2}Cos0\textdegree\\\\\\R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} +2\overrightarrow F_{1}\overrightarrow F_{2}R=\sqrt{({\overrightarrow F_{1} +\overrightarrow F_{2})}^{2}\\R=\overrightarrow F_{1} +\overrightarrow F_{2}\\ R=9 N+7N=16N$as $Cos0\textdegree=1$

Case 2

When both the forces applied in the opposite direction i.e. the angle between the two vector is $180\textdegree$ then they give a resultant minimum as-

$\overrightarrow F_{1} -\overrightarrow F_{2} =9 N-7N=2N$

$R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} +2\overrightarrow F_{1}\overrightarrow F_{2}Cos\theta$

$R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} +2\overrightarrow F_{1}\overrightarrow F_{2}Cos180\textdegree\\R=\sqrt{\overrightarrow F_{1} +\overrightarrow F_{2} -2\overrightarrow F_{1}\overrightarrow F_{2}\\\\R=\sqrt{({\overrightarrow F_{1} -\overrightarrow F_{2})}^{2}\\R=\overrightarrow F_{1} -\overrightarrow F_{2}\\ R=9 N-7N=2N$as $Cos180\textdegree=-1$