Figure P shows how a point on a circle traces a path when it is rolled on the ground. The point in which of the polygons shown in the options creates the path in Figure Q?
Show Hint
The path traced by a rolling shape's point is smooth if the shape is a circle (cycloid) and consists of sharp-cornered arcs if the shape is a polygon. The nature of the arcs depends on the polygon's side lengths and the point's location.
Step 1: Understanding the Concept:
The question explores the concept of a roulette, which is the curve traced by a point attached to a curve as it rolls along another curve.
Figure P shows a cycloid, the path traced by a point on the circumference of a rolling circle. It is a smooth, continuous curve.
Figure Q shows a path made of joined circular arcs, with sharp points (cusps) where the arcs meet and where the path touches the ground. This type of path is generated by a vertex of a rolling polygon.
We need to determine which rolling polygon from the options would generate the path in Figure Q.
Step 2: Detailed Explanation:
Let's analyze the path Q. It consists of identical humps. Each hump starts at the ground, rises along a circular arc, peaks at a cusp, and descends along another circular arc back to the ground. This indicates that the tracing point is a vertex that periodically touches the ground.
Now let's evaluate the options:
Option A (Rectangle): The point is at a vertex. As it rolls, it would trace a sequence of two different circular arcs (one with a radius equal to the short side, one with a radius equal to the long side). The path would not be made of identical repeating humps like in Q.
Option B (Square): The point is at a vertex. The path traced by a vertex of a rolling square is composed of arcs of two different radii (the side length and the diagonal length), which does not match the shape of Q.
Option C (Square): The point is in the middle of a side. This point would never touch the ground. The path Q clearly starts and ends on the ground.
Option D (Parallelogram/Rhomboid): The point is at an obtuse vertex. Let the polygon roll. When the vertex with the blue dot is on the ground, the path starts. As the polygon pivots on the next vertex, the blue dot traces a circular arc whose radius is the side length. As it pivots on the vertex after that, the dot traces a second circular arc, also with the side length as the radius, bringing it back down. This sequence of two identical arcs meeting at a cusp perfectly creates the hump shape shown in path Q.
Step 3: Final Answer:
The path in Figure Q, consisting of repeating humps made of two circular arcs, is generated by the vertex of the rolling parallelogram shown in option D.
