Finding the resultant force on this diagram

Question

I'm trying to find the resultant force on this diagram. However, there's nothing I can see. Also believe that this problem can be done with mathematical calculations.

Regards,

Answer 1

If I read the small picture correctly (all measured in $N$):

• $\vec{F}_1 = \left( \begin{matrix} 4 \\ 0 \end{matrix} \right)$, $\vec{F}_2 = \left( \begin{matrix} 0 \\ 6 \end{matrix} \right)$, $\vec{F}_3 = \left( \begin{matrix} -12 \\ -12 \end{matrix} \right)$
• resulting force: $\vec{F}_r = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = \left( \begin{matrix} -8 \\ -6 \end{matrix} \right)$
• magnitude of resulting force: $|\vec{F}_r|= 10$

Answer 2

In these types of problems you need to choose a set of perpendicular axes, call them $x$ and $y$. They do not necessarily have to be horizontal and vertical. You can decompose any vector into components along $x$ and along $y$, by looking at the angle between a vector and these axes. Do that for every vector. Once you have done that, along each axis you can simply add or subtract real numbers, and you get a resultant component along each axis, call them $R_x$ and $R_y$. The final step is to use Pythagoras' theorem to say $R=\sqrt{R_x^2+R_y^2}$.

In your case, you can choose the axes to be horizontal and vertical. Along the horizontal direction, you have $R_x=+4-12\sqrt{2}\cos 45^\circ$. Write the similar expression for $R_y$, then use the formula that I gave you to calculate $R$

Answer 3

I assume you know complex numbers and how they are presented on the complex plane.

Here, the angle(argument) of a horizontal arrow pointing to the right is $0$.

Then, the arrow pointing upright has an angle of $\frac{\pi }{2}$.

The arrow pointing to bottom left has an angle of $\pi +\frac{\pi}{4}$.

An arrow of length(magnitude) $L$ and angle $\theta$ can be represented by $$Le^{i\theta}$$, where $e$ is the natural e and $i$ is the square root of -1.

The net force equals: $$4e^{0i}+6e^{\pi i/2}+12\sqrt{2}e^{i(\pi +\frac{\pi}{4})}$$.

By using the identity $$e^{i\theta}=\cos(\theta)+i\sin(\theta)$$, I think you are able to obtain an answer in form of $a+bi$.

Its magnitude is $\sqrt{a^2+b^2}$; its angle is $atan2(b,a)$.

Note that $atan2$ is not an ordinary function. Its definition could be found here https://en.m.wikipedia.org/wiki/Atan2