Jaan Kalda kinematics -- Radius of Curvature of a Cycloid

[imbumb]

Cycloid is a curve which can be defined as a trajectory of a point marked on the rim of a rolling wheel or radius R. Determine the curvature radius of such curve at its highest point.

what you need to do is to equate 4v²/r with v²/R and to get that r=4R but i dont understand why the answer shpuldnt just be 2r

 

[topsquark]

Cycloid is a curve which can be defined as a trajectory of a point marked on the rim of a rolling wheel or radius R. Determine the curvature radius of such curve at its highest point.

what you need to do is to equate 4v²/r with v²/R and to get that r=4R but i dont understand why the answer shpuldnt just be 2r

2r is the distance between the highest point and the ground. What they are looking for is the radius of the arc that is made when the point on the wheel is at its highest point. If you look at a picture of it, you will see that the curve made by the point is much flatter than the curve of the wheel.

-Dan

 

[imbumb]

2r is the distance between the highest point and the ground. What they are looking for is the radius of the arc that is made when the point on the wheel is at its highest point. If you look at a picture of it, you will see that the curve made by the point is much flatter than the curve of the wheel.

-Dan

why would the curve of the point be flatter than the curve of the wheel?
edit: i think i get it now

 

[Mister T]

why would the curve of the point be flatter than the curve of the wheel?
edit: i think i get it now

Because the center of curvature is not a fixed location. It moves as the wheel rotates.

 

[A.T.]

Cycloid is a curve which can be defined as a trajectory of a point marked on the rim of a rolling wheel or radius R. Determine the curvature radius of such curve at its highest point.

what you need to do is to equate 4v²/r with v²/R and to get that r=4R but i dont understand why the answer shpuldnt just be 2r

2r is the distance to the instantaneous center of rotation, which equals the curvature radius, if that center of rotation is a static point, but not necessarily if it translates.

 

[vanhees71]

Hm, just do the calculation. Start by parametrizing the cycloid and then analyze it with the usual (plane) Fresnet analysis.

 

[A.T.]

Hm, just do the calculation. Start by parametrizing the cycloid and then analyze it with the usual (plane) Fresnet analysis.

The OP already states a simpler way to calculate it, based on acceleration being the same in the ground frame and the wheel center frame. The question was why the result doesn't match the value intuitively expected by the OP.