Finding the resultant of coplanar forces problem

[Theodore Hodson]

Homework Statement

A uniform ladder of weight W rests with its top against a rough wall and its foot on rough ground which slopes down from the base of the wall at 10 degrees to the horizontal. Resolve, horizontally and vertically, each of the forces acting on the ladder.

Homework Equations

The Attempt at a Solution

Okay so I'm just having some difficulty following this example about coplanar forces from my book.

Looking at the smaller diagrams they drew for splitting R₂ and F₂ into their perpendicular components, I can't seem to figure out how they knew that those angles would be 10 degrees. I know I need to go back to the original diagram and employ some basic geometrical reasoning, starting from the fact that the ground is inclined at 10 degrees. However, I've been fiddling about with this for a while (drawing on the components on the main diagram, forming right triangles, looking for any parallel lines etc) to try and figure out how those angles would be 10 degrees myself but haven't made any headway. So I'm think I'm going about this the wrong way/not noticing something important.

Any hints/tips to help me on this one would be much appreciated :)

 

[Mark44]

Homework Statement

A uniform ladder of weight W rests with its top against a rough wall and its foot on rough ground which slopes down from the base of the wall at 10 degrees to the horizontal. Resolve, horizontally and vertically, each of the forces acting on the ladder.

Homework Equations

The Attempt at a Solution

Okay so I'm just having some difficulty following this example about coplanar forces from my book.

Looking at the smaller diagrams they drew for splitting R₂ and F₂ into their perpendicular components, I can't seem to figure out how they knew that those angles would be 10 degrees. I know I need to go back to the original diagram and employ some basic geometrical reasoning, starting from the fact that the ground is inclined at 10 degrees. However, I've been fiddling about with this for a while (drawing on the components on the main diagram, forming right triangles, looking for any parallel lines etc) to try and figure out how those angles would be 10 degrees myself but haven't made any headway. So I'm think I'm going about this the wrong way/not noticing something important.

Any hints/tips to help me on this one would be much appreciated :)

Turn the first of the two triangles they drew around. $R_2$ is normal to the ladder, so a vertical line starting from the bottom of the ladder makes an angle of 10° with $R_2$. Once you see that, getting the two sides of the triangle are pretty straightforward.

 

[Theodore Hodson]

Turn the first of the two triangles they drew around. $R_2$ is normal to the ladder, so a vertical line starting from the bottom of the ladder makes an angle of 10° with $R_2$. Once you see that, getting the two sides of the triangle are pretty straightforward.

Thanks for the response - I think I can see now how the smaller diagrams fit in with the bigger one. The only thing is I don't get how they knew that the angle between R₂ and the vertical line drawn is 10 degrees. How?

 

[Mark44]

Thanks for the response - I think I can see now how the smaller diagrams fit in with the bigger one. The only thing is I don't get how they knew that the angle between R₂ and the vertical line drawn is 10 degrees. How?

The vertical line is perpendicular to the horizon, right? Through what angle would you have to rotate another vertical line to get one that is perpendicular to the sloping ground? Where you end up is $R_2$, the hypotenuse of that right triangle.

 

[haruspex]

Thanks for the response - I think I can see now how the smaller diagrams fit in with the bigger one. The only thing is I don't get how they knew that the angle between R₂ and the vertical line drawn is 10 degrees. How?

What angle is F₂ to the horizontal? What angle is R₂ to F₂? What angle does that make R₂ to the horizontal?

 

[Theodore Hodson]

What angle is F₂ to the horizontal? What angle is R₂ to F₂? What angle does that make R₂ to the horizontal?

F₂ is 10 degrees to the horizontal. R₂ and F₂ are perpendicular so 90 degrees. So does that mean that R₂ to the horizontal is 100 degrees?

 

[haruspex]

F₂ is 10 degrees to the horizontal. R₂ and F₂ are perpendicular so 90 degrees. So does that mean that R₂ to the horizontal is 100 degrees?

Yes. So what is the angle of R₂ to the vertical?

 

[Theodore Hodson]

Yes. So what is the angle of R₂ to the vertical?

Ahh I think I see now. Given that R₂ is 100 degrees from the horizontal, and drawing in a vertical line perpendicular to the horizontal at the foot of ladder, its clear that the angle R₂ to the vertical will be given by 100-90 .So R₂ to the vertical is 10 degrees and that's where the 10 degrees comes from in the component diagrams. Thank-you very much for the hints :)