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 make use of trigonometry.
First, let's visualize the situation. The cutting tool is a cylindrical disk with a diameter of 80mm and a cutting surface with a radius of 4mm. The surface being produced on the glass lens has a power of -15.00D, and the refractive index of the glass is 1.80.
Now, we need to calculate the angle between the axis of the tool and the axis of the lens. This angle can be found by considering the geometry of the cutting tool and the curvature of the lens surface.
The cutting tool can be thought of as a small section of a larger sphere. The radius of this sphere can be calculated by adding the radius of the cutting surface (4mm) to the radius of the lens surface (-15.00D).
To convert the power to radius, we can use the formula:
P = 1 / f
where P is the power and f is the focal length in meters. Since the power is given in diopters and the radius is in millimeters, we need to convert the units first:
1D = 1 / m
Hence, the focal length of the lens can be calculated as:
f = 1 / (-15.00 D) = -0.067 m
Now, we add the radius of the cutting surface and the radius of the lens surface to find the radius of the larger sphere:
radius of the sphere = 4mm + (-0.067m) = -0.063m
Next, we can calculate the angle between the axis of the tool and the axis of the lens by considering the right triangle formed by the axis of the tool, the axis of the lens, and the radius of the sphere.
The tangent of the angle is given by the equation:
tan(angle) = opposite/adjacent
In this case, the opposite side is 80mm (the diameter of the cutting tool) and the adjacent side is -0.063m (the radius of the sphere).
To use these values, we need to convert the diameter of the cutting tool to millimeters:
diameter = 80mm
Now, we can calculate the angle:
tan(angle) = 80mm / -0.063m
Using the inverse tangent function, we can find the angle:
angle = atan(80mm / -0.063m)
Now, we can calculate the angle using a calculator or a programming language that supports trigonometric functions. The result will be the angle between the axis of the cutting tool and the axis of the lens.