Proving trigonometric identities in a belt and pulley proble

[Terrell]

Homework Statement

verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C]

2. The attempt at a solution
you can see my attempt in the second picture uploaded. i don't think i even got it right

 

[ehild]

Homework Statement

verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C]

2. The attempt at a solution
you can see my attempt in the second picture uploaded. i don't think i even got it right

You are close, but you put A to the wrong place, and c is not the hypotenuse of a right triangle in your picture.
The line x (the connecting belt) makes a right angle with the radius of both circles, and you need to draw a parallel with it from the centre of the smaller circle. You get the yellow rectangle and the green right triangle. Find x and theta from that. Prove both formulas in the OP.

 

[Terrell]

You are close, but you put A to the wrong place, and c is not the hypotenuse of a right triangle in your picture.
The line x (the connecting belt) makes a right angle with the radius of both circles, and you need to draw a parallel with it from the centre of the smaller circle. You get the yellow rectangle and the green right triangle. Find x and theta from that. Prove both formulas in the OP.
View attachment 97665

wow! how did i not see that. thanks! 2(theta) equals to arccosine (D-d)/C right? so to further simplify... theta equals to arccosine[(D-d)/2C] did i got that right?

 

[ehild]

wow! how did i not see that. thanks! 2(theta) equals to arccosine (D-d)/C right? so to further simplify... theta equals to arccosine[(D-d)/2C] did i got that right?

Yes, cos(θ)=(R-r)/c, that is arccos((D-d)/(2c))=θ. But you also have to prove the other formula, for the length of the belt.

 

[Terrell]

one thing i learned is i really have to put all of my thoughts on paper to make things easier. thanks for the help i think i got! check the new image uploaded :)

 

[ehild]

one thing i learned is i really have to put all of my thoughts on paper to make things easier. thanks for the help i think i got! check the new image uploaded :)

It is correct now. And you really need sketches. I am very old and have much experience, but still my first thing is to make a sketch before starting to solve a Physics or Geometry problem.

 

[Terrell]

It is correct now. And you really need sketches. I am very old and have much experience, but still my first thing is to make a sketch before starting to solve a Physics or Geometry problem.

thank you a lot for responding to my thread! i will keep that in mind! :D best of luck