Splitting forces into components

[prune1]

Homework Statement

This problem is actually a worked out solution BUT i don't see where certain things come from...namely how the forces are split into their components. If you look at the attachment, you can see the weight is split into components mgsin and mgcos...I just do not see how this is worked out. I think the axis is being chosen so that the above mgsin and mgCos are parallel and perpendicular to the slope but how do you know which is which? I can't see how this works at all...Please please someone explain to me what weird rotation has gone on here so that the sins and cos are obtained?...

I mean, for example, Fg is acting straight down right, and IT would be the opposite side from where the angle is, so using trigonometry, the component of Fg acting parallel to the slope should be mg/Sin and not mgSin surely?! So confused...

I don't normally have difficulty splitting forces, this one is just odd for some reason and it's put me off completely.

Thanks in advance to ANY who can help!

 

[Nate810]

Homework Equations F = maThe Attempt at a SolutionThe weight of the block is mg, where m is the mass of the block and g is the acceleration of gravity. The force of gravity can be split into two components: mgsinθ and mgcosθ. The mgsinθ component is parallel to the slope and the mgcosθ component is perpendicular to the slope.

 

[ogb p]

The process of splitting forces into components involves breaking down a single force into its individual components along different axes. This is done in order to simplify the analysis of the forces acting on an object in a particular direction. In the attachment provided, the weight (Fg) is split into two components, mgcosθ and mgsinθ, where θ is the angle of the slope.

To understand how this is done, let's first consider the definition of the sine and cosine functions. In a right-angled triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. Similarly, the cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse.

Now, in the case of the weight (Fg), it is acting straight down and is opposite the angle θ. This means that the opposite side of the triangle is mg and the hypotenuse is Fg. Using the definition of sine, we can write:

sinθ = mg/Fg

Rearranging this equation, we get:

mg = Fg sinθ

This means that the component of Fg acting parallel to the slope (mgsinθ) is equal to Fg multiplied by the sine of the angle. Similarly, the component of Fg acting perpendicular to the slope (mgcosθ) is equal to Fg multiplied by the cosine of the angle.

In summary, the process of splitting forces into components involves using trigonometric functions to find the individual components of a force along different axes. In this case, the axis is chosen so that the components are parallel and perpendicular to the slope. I hope this explanation helps to clarify the process for you.