A spherical generator is used to produce a -15.00D surface on glass of refractive index 1.80. The diameter of the cutting tool is 80mm and the radius of the cutting surface is 4mm. What is the angle between the axis of the tool and the axis of the lens?
To find the angle between the axis of the tool and the axis of the lens, we need to use some basic trigonometry.
First, let's visualize the situation. The spherical generator is shaping a -15.00D surface on the glass, and the cutting tool has a diameter of 80mm. We can assume that the axis of the tool is aligned with the axis of the lens.
We can use the radius of the cutting surface (4mm) to determine the distance from the center of the cutting surface to the edge of the tool. This distance is the radius of the tool (40mm) plus the radius of the cutting surface (4mm), which equals 44mm.
Next, we can use the diameter of the cutting tool (80mm) to determine the distance between the center of the tool and the edge of the tool. This distance is half the diameter of the tool, which is 40mm.
Now, using these two distances, we can create a right triangle. The vertical leg of the triangle represents the distance from the center of the cutting surface to the edge of the tool (44mm), and the horizontal leg represents the distance from the center of the tool to the edge of the tool (40mm).
We can use the tangent function to find the angle between the axis of the tool and the axis of the lens. The tangent of an angle is equal to the ratio of the length of the opposite side (vertical leg in this case) to the length of the adjacent side (horizontal leg).
Let's calculate it:
tan(theta) = (44mm / 40mm)
tan(theta) = 1.1
To find the value of theta, we can take the inverse tangent (arctan) of 1.1 using a calculator or lookup table.
theta ≈ 47.13 degrees
So, the angle between the axis of the tool and the axis of the lens is approximately 47.13 degrees.