A wheel is performing pure rolling on the horizontal surface with constant velocity V the radius of the wheel is R . Find the point on the circumference , where the radius of curvature is equal to R

To find the point on the circumference of a wheel performing pure rolling on a horizontal surface where the radius of curvature is equal to the radius of the wheel (R), we need to delve into some concepts of geometry and motion. Let's break this down step by step.

Understanding the Wheel's Motion

When a wheel rolls without slipping, every point on the circumference of the wheel traces a path known as a cycloid. The wheel maintains a constant velocity, which means that the center of the wheel moves forward at a steady speed, while the points on the circumference have different velocities depending on their position.

Radius of Curvature Explained

The radius of curvature at a point on a curve is a measure of how sharply the curve bends at that point. For a circle, the radius of curvature is constant and equal to the radius of the circle. However, for a point on the circumference of a rolling wheel, the radius of curvature can vary depending on the position of that point.

Identifying the Relevant Point

To find the specific point on the circumference where the radius of curvature equals R, we can consider the geometry of the wheel. The wheel's center is at a height of R above the ground, and as it rolls, the points on the circumference will have different curvatures based on their position relative to the ground.

Analyzing the Circumference

• The bottommost point of the wheel, which is in contact with the ground, has a radius of curvature that is effectively zero because it is a point of tangency.
• The topmost point of the wheel, which is the highest point above the ground, has a radius of curvature that is equal to R, as it is part of the circular arc of the wheel.
• Points on the sides of the wheel will have varying radii of curvature that are less than R.

Conclusion on the Point of Interest

Therefore, the point on the circumference of the wheel where the radius of curvature equals R is the topmost point of the wheel. This point is directly above the center of the wheel when it is in motion. In summary, when the wheel rolls with a constant velocity V, the point on the circumference where the radius of curvature is equal to the radius of the wheel R is located at the highest point of the wheel.