Short Answer:

A cycloid is a curve traced by a point on the rim of a circle as it rolls along a straight line without slipping. This curve is commonly used in gear profiles and cam design. Drawing a cycloid involves dividing the rolling circle and the path into equal parts, locating corresponding points, and joining them smoothly.

It is an important part of engineering graphics and is useful in understanding how rotating objects move. With simple steps using compass and scale, you can draw a cycloid accurately for both learning and practical use in mechanical drawings.

Detailed Explanation:

Procedure for drawing a cycloid

A cycloid is a type of curve seen in many mechanical applications, especially in the design of gears, cams, and mechanisms where rolling motion is involved. It is defined as the path followed by a fixed point on the circumference of a circle as it rolls along a straight horizontal line without slipping.

This curve is not only important theoretically but also helps students and engineers visualize how rotation and translation combine during motion. Below is a step-by-step procedure to draw a cycloid accurately on paper using basic drawing instruments.

Step-by-step procedure

Step 1: Draw the base line and circle

Draw a horizontal straight line which represents the path or base line along which the circle will roll.
Draw a circle with a chosen radius (say 30 mm). This is the rolling circle.
Mark the starting point of the circle where the curve will begin.

Step 2: Calculate the total length of the base

The length of one complete revolution of the circle (called the pitch of the cycloid) is equal to the circumference of the circle.
Use the formula:
Circumference = 2 × π × r
For a radius of 30 mm, circumference ≈ 188.4 mm.
Mark this length as a straight horizontal line below the circle.

Step 3: Divide the circle and base line

Divide the circle into equal parts, usually 12 or 8 divisions, along its circumference.
Similarly, divide the base line into the same number of equal parts.
Label the points on both the circle and the base line as 1, 2, 3, and so on.

Step 4: Locate the points on the cycloid

For each division:
- Imagine the circle rolling forward without slipping.
- At each step, the center of the circle moves to the corresponding point on the base.
- The point on the circle that traces the cycloid rotates down as the circle rolls.
To locate each cycloid point:
- From each base point, draw a vertical line upward.
- Using a compass, take the radius of the circle.
- From each new circle center (on the base), rotate the point from the circle’s edge to its new position using an angle equal to the arc traveled.
- This process is repeated for all points.

Step 5: Mark and join the curve

After marking all the points traced by the rotating point, join them smoothly using a French curve or freehand.
This final curve is the cycloid.

Notes for accuracy

The more divisions you take (like 12 or 24), the smoother and more accurate your cycloid will be.
Always keep the arc distance equal to base distance for each division.
Ensure the circle rolls without slipping—this is the basic condition for a cycloid.

Applications in mechanical engineering

Cycloids are used in gear teeth design (especially cycloidal gears).
They are important in cam profile design, where smooth and accurate motion is needed.
Cycloidal paths are also studied in mathematical modeling of rolling motion.

Understanding how to draw a cycloid manually helps in understanding motion curves, which is a part of kinematics and mechanism design in mechanical engineering.

Conclusion:

Drawing a cycloid involves rolling a circle over a straight line and tracing the path of a fixed point on its circumference. The procedure includes dividing the circle and base line, locating key points using compass and vertical lines, and joining them to form the curve. Cycloids have practical importance in gears and cams, and learning to draw them builds strong understanding in engineering graphics and mechanical motion design.