A circle of 115 mm diameter rolls on another circle of 75 mm diameter with internal contact. Draw the locus of a point on the circumference of the rolling circle for its one complete revolution.

To draw the locus of a point on the circumference of the rolling circle for its one complete revolution, we can use the concept of cycloids.

A cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line. In this case, the rolling circle has a diameter of 115 mm and the fixed circle has a diameter of 75 mm.

To draw the locus of a point on the rolling circle, we can first draw the fixed circle with a diameter of 75 mm. Next, we draw a horizontal line that is tangent to the fixed circle at its lowest point. This line represents the straight line along which the rolling circle will roll.

Now, we can draw the rolling circle with a diameter of 115 mm. We mark a point on its circumference and trace its path as it rolls along the horizontal line. The locus of this point will be a cycloid.

To complete one full revolution of the rolling circle, we need to trace the path of the point on the rolling circle for a distance equal to the circumference of the rolling circle. This will give us the complete cycloid curve.

The final drawing will show the cycloid curve traced by a point on the rolling circle as it completes one full revolution.

Steps to draw the locus:

1. Draw the initial position of the rolling circle: Draw a circle of 115 mm diameter and mark a point P on its circumference.
2. Draw the path of the center of the rolling circle: Draw a circle of 75 mm diameter that will form the base for the rolling circle.
3. Roll the circle without slipping: As the circle rolls, point P will trace a path around the base circle.
4. Mark the locus of point P: As the circle rolls through one complete revolution, point P will trace a curve on the base circle. This curve is the locus.

Characteristics of the locus:

- Curve: Cycloid
- Equation:
```
x = a(θ - sin θ)
y = a(1 - cos θ)
```
where:
- a is the radius of the rolling circle (57.5 mm)
- θ is the angle of rotation of the rolling circle

Drawing the locus:

Using the equation of the cycloid, we can plot points and connect them to form the locus. Here's how to do it:

1. Calculate the coordinates for several values of θ (e.g., θ = 0, π/4, π/2, 3π/4, π).
2. Plot these points on the base circle.
3. Connect the points with a smooth curve to create the locus.

The resulting locus will be a curve with the shape of an inverted arch.