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 can use trigonometry. We'll first calculate the height of the lens or the distance between the axis of the lens and the cutting surface.
Given:
Refractive index of glass (n) = 1.80
Diameter of the cutting tool (D) = 80mm
Radius of the cutting surface (r) = 4mm
Desired surface power (P) = -15.00D
The power of the lens is given by the formula:
P = (n - 1) / r
Rearranging the formula, we have:
r = (n - 1) / P
Plugging in the given values, we have:
r = (1.80 - 1) / -15.00D
Simplifying, we get:
r = -0.8 / -15.00D
r = 0.0533mm
Now, we can find the height of the lens using the radius:
Height (h) = sqrt((D/2)^2 - r^2)
Plugging in the values, we have:
h = sqrt((40mm)^2 - (0.0533mm)^2)
Simplifying, we get:
h = sqrt(1600mm^2 - 0.0028mm^2)
h = sqrt(1599.9972)mm
Now, we have the height of the lens. To find the angle between the axis of the tool and the axis of the lens, we can consider the right triangle formed by the height, radius, and the angle.
Let the angle be θ.
We have:
tan(θ) = r / h
Plugging in the values, we get:
tan(θ) = 0.0533mm / sqrt(1599.9972)mm
Calculating θ using a calculator, we find:
θ = 0.001058 radians
To convert this to degrees, multiply by 180/π:
θ = 0.001058 * (180/π)
Therefore, the angle between the axis of the tool and the axis of the lens is approximately 0.061 degrees.