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.imbumb said: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
why would the curve of the point be flatter than the curve of the wheel?topsquark said: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
Because the center of curvature is not a fixed location. It moves as the wheel rotates.imbumb said:why would the curve of the point be flatter than the curve of the wheel?
edit: i think i get it now
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.imbumb said: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
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.vanhees71 said:Hm, just do the calculation. Start by parametrizing the cycloid and then analyze it with the usual (plane) Fresnet analysis.
A cycloid is a curve traced by a point on the circumference of a circle as the circle rolls along a straight line. It is a specific type of trochoid, which is a curve generated by a point on the circumference of a circle as it rolls along a fixed circle.
Jaan Kalda is a physicist and mathematician from Estonia who is known for his contributions to the field of kinematics. He is associated with the concept of the radius of curvature of a cycloid because he developed a mathematical formula for calculating this value for any point on the curve.
The radius of curvature of a cycloid is the radius of the osculating circle at a specific point on the curve. In other words, it is the radius of the circle that best approximates the curve at that point.
The radius of curvature of a cycloid can be calculated using the formula R = a(1 + cosθ), where R is the radius of curvature, a is the radius of the generating circle, and θ is the angle between the tangent to the curve and the horizontal line passing through the point of interest.
The radius of curvature of a cycloid has applications in engineering, physics, and mathematics. It is used in the design of gears and other mechanical systems, as well as in the analysis of motion and forces in various systems. It also has applications in optics and the study of light and sound waves.