Question

Calculate the angle between a $2N$ and $3N$ force such that their resultant is $4N$.

Answer 1

Hint:Use the equation for addition of two vectors. All the data except the angle are given. Put everything and calculate the angle.The subtraction of two vectors is calculated using a similar vector with a minor change in sign.

Complete step by step answer:
Let ${F_1} = 2N$ and ${F_2} = 3N$
Given the resultant of two forces is $4N$.
Let the required angle be $\theta $.
Resultant means the addition of two vectors.
Here since only two vectors are given, there is no need to resolve into it’s a axis and y axis components. We can directly apply the below formula.
Thus resultant of the two vectors is given by \[R = \sqrt {{F_1}^2 + {F_2}^2 + 2{F_1}{F_2}\cos \theta } \]
Now we put all the given data in the equation
\[
4 = \sqrt {{3^2} + {2^2} + 2 \times 2 \times 3\cos \theta } \\
\Rightarrow 4 = \sqrt {13 + 2 \times 2 \times 3\cos \theta } \\
\Rightarrow 13 + 2 \times 2 \times 3\cos \theta = 16 \\
\Rightarrow 12\cos \theta = 3 \\
\Rightarrow \cos \theta = \dfrac{1}{4} \\
\therefore\theta = {\cos ^{ - 1}}\dfrac{1}{4} \\
\]
Finally we get that the angle is \[\theta = {\cos ^{ - 1}}\dfrac{1}{4}\].

Additional Information:
The subtraction of two vectors is given by
\[R = \sqrt {{F_1}^2 + {F_2}^2 - 2{F_1}{F_2}\cos \theta } \]
To find the resultant of vectors, there are three laws of addition namely the triangle law, the parallelogram law and the polygon law. The triangle and parallelogram law are used for only two vectors. But the polygon law can be used for any number of vectors. These laws are very useful in calculations in vector physics.

Note:Students should know the formula for the resultant of two vectors. Also sometimes instead of two vectors, more vectors can be given. For this the resultant is more cleverly calculated by resolving down each force into its x axis and y axis component and then calculating the resultant.