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 geometric principles.
First, let's visualize the problem. We have a spherical generator that is used to cut a -15.00D (diopters) surface on a glass of refractive index 1.80. The cutting tool has a diameter of 80mm, and the radius of the cutting surface is 4mm.
To calculate the angle, we can use trigonometry. The key is to find the height difference (h) between the center of the tool and the center of the spherical surface. Then, we can use the tangent function to find the angle.
Here's how we can proceed step-by-step:
1. Find the height difference (h):
- The height difference (h) can be calculated using the following formula:
h = radius - (radius - tool diameter/2)
- Plugging in the values, we get:
h = 4mm - (4mm - 80mm/2)
h = 4mm - (4mm - 40mm)
h = 4mm - (-36mm)
h = 40mm
2. Calculate the angle:
- Now that we have the height difference (h), we can use the tangent function to find the angle (θ):
tan(θ) = h/working distance
- The working distance is given by the formula:
working distance = lens diameter/2 - radius
- Plugging in the values, we get:
working distance = 80mm/2 - 4mm
working distance = 40mm - 4mm
working distance = 36mm
- Now, substitute the values of h and working distance into the tangent function:
tan(θ) = 40mm/36mm
- Finally, solve for θ by taking the inverse tangent (arctan) of both sides:
θ ≈ arctan(40mm/36mm)
Using a scientific calculator, we find that:
θ ≈ arctan(40/36)
θ ≈ arctan(1.1111)
θ ≈ 46.098°
Therefore, the angle between the axis of the tool and the axis of the lens is approximately 46.098°.