# Math Help - Magnitude and Direction

1. ## Magnitude and Direction

Hey guys,

This is more for the Physics forum, but there's noone over there.

Three workers are pulling on ropes attached to a tree stump (S) as shown in the diagram.
Find the magnitude and direction of the resultant force on the stump.

Diagram is attached at the bottom.

A force of 60 Newtons North, a force of 35 Newtons East and a force of 40 Newtons 60 degrees South of East is applied.

I know the answer is 61 Newtons at a direction of 23 degrees North of East, but I'll be danged if I can get there.

A little nudge in the right direction would be helpful.

2. Make a polygon of the forces and then use the trigonometric functions to solve the distance.

3. Revised drawing:

I have two resultant vectors with two different magnitudes with 2 different directions.

I need these two forces to combine to 61 Newtons somehow. Do I combine those two vectors and find the resultant of these two forces?

"My mind's going, Dave. I can feel it."

4. Decompose the force vectors into each of their x- and y-components, then sum the x-components and sum the y-components. The resulting x- and y-components are the components of the resultant force.

Note the x- and y-components of a force vector $\vec A$ is given by

$x = |\vec A|\cos \theta$
$y = |\vec A|\sin \theta$

where $|\vec A|$ is the magnitude of the force vector and $\theta$ is the angle between the force vector and the x-axis.

5. It is not need to make two result vectors and then find final resultant. Just make a polygon of vectors and use trigonometric. Find what is a polygon of the vectors. This way is better because when you will get 10, 15 vectors, It is not need two make resultant for each two vectors.

Kind regards

6. Originally Posted by masters
Revised drawing:

I have two resultant vectors with two different magnitudes with 2 different directions.

...
If you want to use both resultant forces you would have used the force of 35 N twice!

Originally Posted by masters
Hey guys,

Three workers are pulling on ropes attached to a tree stump (S) as shown in the diagram.
Find the magnitude and direction of the resultant force on the stump.

Diagram is attached at the bottom.

...
I assume that you are looking for a more geometric way to solve this question:

1. I've attached a (nearly exact) drawing. Scale is 1 cm correspond to 10 N.

2. Calculate the green resultant first using Pythagorean theorem ( $\sqrt{4825}$)

2. Calculate the green angle: $\tan^{-1}\left(\frac{60}{35}\right)\approx 59.744^\circ$

3. Use the greyed triangle and the Cosine rule to calculate the final (red) resultant:

$R = \sqrt{4825 + 1600 - 2 \cdot \sqrt{4825} \cdot 40 \cdot \cos(60.256^\circ)}$

I've got $R \approx 60.565\ N$

4. Use Sine rule or Cosine rule to calculate the missing angle.