# Statics: two unknown angles and resultant force

• drnoisewater1
In summary, two forces of 100N and 200N are applied to the upper corner of a crate. The resultant vector, R, has a magnitude of 250N and is directed to the right (+x direction).
drnoisewater1

## Homework Statement

Two forces P and Q with respective magnitudes 100N and 200N are applied to the upper corner of a crate. The sum of the two forces is to the right (+x direction) with a magnitude of 250N. Find the angles that P and Q make with their sum - that is, with the horizontal line through +x axis.

## Homework Equations

R = P + Q where R is the resultant vector and P and Q are vectors given in the problem

Rx = Px + Qx = 250

Ry = Py + Qy = 0

## The Attempt at a Solution

All the information is given except the two angles. Plugging in the given values gives me the equation:

$$\Sigma$$Rx = 100cos($$\theta$$) + 200cos($$\phi$$) = 250N

$$\Sigma$$Ry = 100sin($$\theta$$) - 200sin($$\phi$$) = 0

where theta is the angle between P and R and phi is the angle between Q and R.

Basically this comes down to confusion on algebra for me. I attempted substitution and that got me nowhere. I know there is some kind of trick to solving this, but I cannot figure it out. There are two equations and two unknowns so there must be a way to do it. I have worked a similar problem where one of the angles is known and the other was supposed to be at a maximum and calculus could be used there. Is that a possibility in this problem or is there just a little trick that I am missing?

How about $$tan(\theta)$$ does that equal anything?

I tried that, but still had phi in the expression.

Rx: sin($$\theta$$) = 2sin($$\phi$$)

Ry: 2.5 - 2cos($$\phi$$)

I thought that since magnitude of R is sqrt[Rx^2 + Ry^2] = 250 things might cancel out. Well all the angles canceled out so i just got an incorrect expression.

I tried Ry/Rx just for fun, to get a tan expression and that was not beneficial because nothing canceled out there either. I just thought about this though: if you take tan inverse of Ry/Rx the resultant angle will be 0 since the resultant is about the x-axis. However, that means little as I don't know if you can evaluate arctan[0.8sin$$\phi$$) - tan($$\phi$$)]

Since vectors add head-to-tail, could you maybe draw a force triangle with P, Q, and R and find the angles that way?

## What is statics and why is it important?

Statics is a branch of mechanics that deals with the analysis of forces acting on a stationary object. It is important because it helps us understand how objects behave under different forces and how to design structures that can withstand these forces.

## What are two unknown angles in statics?

In statics, there can be two unknown angles: the angle of the resultant force and the angle between two forces. The angle of the resultant force is the angle between the resultant force and the horizontal axis, while the angle between two forces is the angle between the two forces acting on an object.

## How do you calculate the resultant force in statics?

To calculate the resultant force in statics, you need to use vector addition. This involves breaking down all the forces acting on an object into their horizontal and vertical components, then adding these components together using vector addition. The magnitude and direction of the resultant force can then be determined from the resulting vector.

## What is the principle of moments in statics?

The principle of moments states that for an object to be in equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments. This means that the object will not rotate as long as the moments on each side are balanced.

## How can you solve for unknown angles and resultant force in statics?

To solve for unknown angles and resultant force in statics, you need to use the principles of vector addition, equilibrium, and moments. By setting up equations using these principles, you can solve for the unknown angles and resultant force using basic algebraic operations.

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