Question

If $N$ is resolved in two components such that the first is twice of other, the components are

• A. $5N,5\sqrt{2}N$
• B. $10N,10\sqrt{2}N$
• C. $\frac{N}{\sqrt{5}},\frac{2N}{\sqrt{5}}$
• D. None of these

Answer 1

The correct option is C

$\frac{N}{\sqrt{5}},\frac{2N}{\sqrt{5}}$

Step 1: Given data

$N$ is resolved into two components such that the first is twice of the other.

We have to find the two components.

Step 2: Formula and solution

Let the two components be $N_x$ and $N_y$ such that $N_x=N\cos A$ and $N_y=N\sin A$

Given, one component is twice the other.

Therefore,$N_x=2N_y$

Since $N_x$ and $N_y$ are components of vector $N$,

$$\begin{gathered} N_x^2+N_y^2=N^2 \\ {\left(2N_y\right)}^2+N_y^2=N^2\left(\because N_x=2N_y\right) \\ 4N_y^2+N_y^2=N^2 \\ 5N_y^2=N^2 \\ \Rightarrow N_y=\frac{N}{\sqrt{5}} \\ \Rightarrow N_x=2N_y=\frac{2N}{\sqrt{5}} \end{gathered}$$

We get the value of components as $\frac{N}{\sqrt{5}},\frac{2N}{\sqrt{5}}$

Therefore, Option C is correct.