Why is the triangle edge length Rθ in circular motion?

[MienTommy]

That's the image. I can't see how the edge of that middle triangle's length is Rθ at all. I've tried using similar triangles and law of cosines to figure that out but I'm missing lengths. I thought Rθ was the arc length of a piece of a circle? So if Rθ is a length of a piece of a circle, how is Rθ equal to that edge of the triangle?

 

[insightful]

I can't see how the edge of that middle triangle's length is Rθ at all.

It looks to me like this is a "given." You might be seeing right angles where none exist.

 

[CWatters]

Took me a few moments to figure it out. The answer is that the block started off horizontal then rocks to the left without slipping on the cylinder ... so the red and green lines in this version must be the same length...

 

[insightful]

Took me a few moments to figure it out. The answer is that the block started off horizontal then rocks to the left without slipping on the cylinder ... so the red and green lines in this version must be the same length...

View attachment 93508

But your green line doesn't equal R*theta as labeled, right?

 

[insightful]

But your green line doesn't equal R*theta as labeled, right?

I.e., the extension of the radius R to the mid-point line of the bar is not parallel to the line from the top of the cylinder to the big black dot.

 

[CWatters]

Why not?

 

[CWatters]

I had another think about this. I think you are right. I think the bit in yellow should have been omitted.

It becomes obvious something is wrong when theta is 90 degrees.

 

[insightful]

It becomes obvious something is wrong when theta is 90 degrees.

Bingo; that was my "aha" moment also.