Angle of Tension b/w 2 Dogs Pulling Ropes?

[Nano-Passion]

Homework Statement

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60 degrees. If dog A exerts a force of 270N and dog B exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with dog A's rope.

Homework Equations

The Attempt at a Solution

We know that the resultant force between 2 forces is

$$f_{res} = \sqrt{F_{a}^2 + F_{b}^2 + 2 F_{a} F_{b} cos \theta}$$
$= 493.86N$

This part I got write, the part I don't understand is how to find the angle the resultant force makes with the dog A's rope.

I grabbed the problem on cramster http://www.cramster.com/solution/solution/1339563 and the person uses this formula

$$tan \theta = \frac{F_{b} sin 60}{F_{a} + F_{b} cos 60}$$

Homework Statement

But after 30min+ grappling with how he got to that formula I still don't know how!

Please help, I'm going crazy over this lol.

 

[grzz]

I prefer to use components rather some formula.

But since you used the cosine formula for the resultant you can use the sine formula to get the angle:

300/sintheta = resultant/sin120

 

[Nano-Passion]

I prefer to use components rather some formula.

But since you used the cosine formula for the resultant you can use the sine formula to get the angle:

300/sintheta = resultant/sin120

You can't because they only give you the angle between the rope, not the direction of pull.

How did you get sin 120?

 

[grzz]

If you look at the pararellogram whose sides are the tensions, one can use the sine formula in the traingle made up by the resultant and the two tensions. The angle opposite the resultant is 120deg.

 

[asteriatic]

I would approach this problem by first drawing a free body diagram of the situation. This will help us visualize the forces acting on the post and the direction of the resultant force.

In this case, we have two forces acting on the post - one from dog A and one from dog B. These forces are acting at different angles, so we can break them down into their x and y components.

Using trigonometry, we can find the x and y components of the forces:

F_{ax} = F_{a} cos 60 = 135 N
F_{ay} = F_{a} sin 60 = 233.55 N
F_{bx} = F_{b} cos 60 = 150 N
F_{by} = F_{b} sin 60 = 259.81 N

Now, we can find the resultant force by adding the x and y components:

F_{rx} = F_{ax} + F_{bx} = 285 N
F_{ry} = F_{ay} + F_{by} = 493.86 N

The magnitude of the resultant force can be found using Pythagorean theorem:

F_{res} = \sqrt{F_{rx}^2 + F_{ry}^2}
= 568.07 N

To find the angle the resultant force makes with dog A's rope, we can use inverse trigonometric functions:

tan \theta = \frac{F_{ry}}{F_{rx}}
\theta = tan^{-1} \left(\frac{F_{ry}}{F_{rx}}\right)
= tan^{-1} \left(\frac{493.86}{285}\right)
= 59.2^{\circ}

So, the angle between the resultant force and dog A's rope is 59.2 degrees.

As for the formula used on Cramster, it is based on the definition of tangent function and the fact that the sum of all angles in a triangle is 180 degrees. Using this formula, we can also find the same result:

tan \theta = \frac{F_{by}}{F_{bx} + F_{ax}}
= \frac{259.81}{150 + 135}
= 59.2^{\circ}

I hope this helps clarify the solution for you. As a scientist, it is important to understand the underlying principles and concepts