Question

The curve traced by a point on a circle rolling inside another circle is known as

• A. hypocycloid.
• B. helix.
• C. involute.
• D. hyperbola.

Hint:

Hypocycloids occur when a circle rolls inside another circle, producing sharp points at regular intervals depending on the size ratio between the two circles.

Answer 1

The Correct Option is C

The curve traced by a point on a circle rolling inside another circle is called a hypocycloid. A hypocycloid is formed when a smaller circle rolls inside a larger fixed circle, and the point of interest traces a path on the circumference of the smaller circle. The resulting curve has characteristic cusp points, depending on the size ratio between the two circles.

Step 1: Explanation of hypocycloid.
A hypocycloid is defined mathematically as the curve traced by a point on a circle as it rolls without slipping inside a larger circle. The key characteristic is that the point moves along a specific path that has sharp points or cusps, unlike a smooth curve such as a circle. The equation for a hypocycloid in polar coordinates is derived from geometric properties of the rolling circle.

Step 2: Comparing with other options.
- (B) Helix: A helix is a three-dimensional spiral curve, not related to rolling circles.
- (C) Involute: An involute is a curve traced by a point on a string as it is unwound from another curve, not related to a rolling circle.
- (D) Hyperbola: A hyperbola is a type of conic curve, unrelated to the process of rolling circles.
Thus, the correct answer is a hypocycloid, which is a type of curve that can occur in the study of gears and other mechanical applications involving rolling circles.

Final Answer: (A)