A cicular opaque disk, 25cm in diameter, is set up 30 cm from a circular source of light of diameter 40 cm. A screen is placed 30cm beyond the opaque disk. Describe, giving dimensions, the nature of the shadow on the screen.
In all the time you've spent posting this problem, have you tried studying your text, or heck, just thinking about it?
Draw a diagram of the scene seen edge-on.
Let the line MN representing the edge of the mirror extend from M=(0,20) to N=(0,-20)
Let the line AB representing the disc extend from A=(30,12.5) to B=(30,-12.5)
Let the screen be drawn at x=60
Extend line MA to V on the screen
Extend line MB to W
Extend line NA to X
Extend line NB to Y
The line VX represents the diameter of the umbra, since it is in the shadow of all light points between M and N
Lines VY and XW represent the ring of the penumbra, since they are illuminated by some rays from the light.
A little algebra reveals that
XV = 15.625
VY = 45 - 15.625 = 29.375
So, the umbra is a disc of diameter 31.25cm
the penumbra is a ring around the umbra, with diameter 90cm
what is the algebra to reveal
xv=15.625?
trying to get my head around this but it is not sinking in
Actually, I messed up.
M=(0,20) A = (30,12.5)
The line MA has slope -1/4 (-7.5/30)
So, XV = 5
The line NA has slope 32.5/30 = 65/60
So, Y is (60,45), and VY = 40
this changes the diameter of the umbra, but not the penumbra.
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To describe the nature of the shadow on the screen, let's break it down step by step:
1. The circular source of light has a diameter of 40 cm. This means that light is emitted from all parts of the source with a maximum radius of 20 cm (half of the diameter).
2. The opaque disk is placed 30 cm away from the source of light. Since the diameter of the disk is 25 cm, the disk is smaller than the source of light. This means that some of the light from the source will pass around the disk and continue towards the screen.
3. We also know that the screen is placed 30 cm beyond the opaque disk. This means that the distance between the source of light and the screen is 60 cm (30 cm from the disk + 30 cm from the disk to the screen).
Based on these dimensions, we can describe the nature of the shadow on the screen:
a) The shadow on the screen will have a circular shape. This is because both the source of light and the opaque disk are circular.
b) The shadow will have a bright central region surrounded by a darker outer region. The bright central region is the portion of light that passes through the gap between the disk and the screen. The darker outer region is the area where the opaque disk blocks the direct light from reaching the screen.
c) The size of the bright central region will be approximately 40 cm in diameter. This is the diameter of the source of light, and since the disk is smaller, the entire light emitted from the source will pass through the gap.
d) The larger diameter shadowed area on the screen will be approximately 25 cm in diameter. This is the size of the opaque disk, and it will block the direct light from reaching the screen, resulting in a shadowed region.
So, in summary, the nature of the shadow on the screen will be a circular shape with a bright central region of approximately 40 cm in diameter and a darker outer region surrounded by a shadowed area of approximately 25 cm in diameter.