The moon's diameter is 3.48 106 m, and its mean distance from the earth is 3.85 108 m. The moon is being photographed by a camera whose lens has a focal length of 48.5 mm.
(a) Find the diameter of the moon's image on the slide film.
(b) When the slide is projected onto a screen that is 15.1 m from the lens of the projector (f = 103.4 mm), what is the diameter of the moon's image on the screen?
I got part a to be 4.38e-4 m but have no idea how to do b
(a)
1/do+1/di =1/F
1/di=1/F-1/do=1/48.5•10⁻³- 1/3.85•10⁸.
di≃48.5•10⁻³m
x/di=D/do
x=D•di/do =3.48•10⁶•48.5•10⁻³/3.85•10⁸.=
=4.38•10⁻⁴ m
(b)
1/do₁+1/di₁ =1/f
1/do₁=1/f- 1/di₁ =
=1/103.4•10⁻³-1/15.1 ≃ 9.6
do₁=0.1404 m
x/do₁=Y/di₁
Y=di₁•x/do₁=15.1•4.38•10⁻⁴/0.104 = 0.0636 m
Why did the moon go to the therapist?
Because it had projection issues!
Alright, let's try to solve part (b) together. To find the diameter of the moon's image on the screen, we can use the thin lens formula:
1/f = 1/v - 1/u
Where:
f is the focal length of the projector lens
v is the distance between the lens and the image (on the screen)
u is the distance between the lens and the object (on the slide film)
Now, since we already know the focal length (f = 103.4 mm) and we need to find the diameter of the moon's image on the screen, let's assume the image is formed at the focal point of the lens (v = f).
Using the thin lens formula, we can rearrange it to solve for u:
1/u = 1/f - 1/v
Plugging in the values we know:
1/u = 1/103.4 mm - 1/103.4 mm (since v = f)
Simplifying:
1/u = 0
Uh-oh! It seems we have encountered a mathematical joke called a "divide by zero" error. This means that the image formed by the lens is infinitely far away from the lens, and therefore, we can't determine its size on the screen.
Looks like the moon's image decided to pull a disappearing act on us! Apologies if my answer wasn't as illuminating as the projector screen. Feel free to reach out if you have any other questions, or if you just need another silly joke!
To find the diameter of the moon's image on the screen (part b), we can use the thin lens formula:
1/f = 1/do + 1/di
Where:
- f is the focal length of the lens (103.4 mm)
- do is the object distance (15.1 m)
- di is the image distance (unknown)
First, we need to convert the focal length and object distance to meters:
f = 103.4 mm = 0.1034 m
do = 15.1 m
Now, we can rearrange the formula to solve for di:
1/di = 1/f - 1/do
1/di = 1/0.1034 - 1/15.1
To simplify this, we need to find a common denominator:
1/di = (15.1 - 0.1034) / (0.1034 * 15.1)
1/di = 15.0066 / 1.55834
Now, we can find the value of di:
1/di ≈ 9.63
di ≈ 0.1039 m
So, the image distance is approximately 0.1039 meters.
Now, to find the diameter of the moon's image on the screen, we can use the magnification formula:
magnification = - di / do
Where:
- di is the image distance (0.1039 m)
- do is the object distance (15.1 m)
magnification = - 0.1039 / 15.1
magnification ≈ -0.00688
Since the magnification is negative, it means the image is inverted. We can ignore the negative sign when finding the diameter.
The diameter of the moon's image on the screen is:
diameter = (moon's diameter) * magnification
diameter ≈ 3.48 × 10^6 m * 0.00688
diameter ≈ 23,971.2 m
Therefore, the diameter of the moon's image on the screen is approximately 23,971.2 meters.
To solve part b of the question, we can use the thin lens formula and the concept of similar triangles. Here is the step-by-step explanation:
Step 1: Convert the given measurements to meters to maintain consistency.
- The focal length of the lens is 48.5 mm = 48.5 × 10^-3 m.
- The distance between the lens of the projector and the screen is 15.1 m.
- The diameter of the moon's image on the slide film, calculated in part a, is 4.38 × 10^-4 m.
Step 2: Apply the thin lens formula, which relates the object distance (do), image distance (di), and the focal length (f) of a lens:
1/f = 1/do + 1/di
Step 3: Rearrange the thin lens formula to solve for the image distance (di) on the screen:
1/di = 1/f - 1/do
Step 4: Substitute the given values into the equation:
1/di = 1/(103.4 × 10^-3) - 1/(4.38 × 10^-4)
Step 5: Calculate the value of (1/di):
1/di ≈ 9684.99 - 228310.81 ≈ -218625.82
Step 6: Take the reciprocal of (1/di) to determine the image distance (di) on the screen:
di ≈ 1/(-218625.82) ≈ -4.57 × 10^-6 m
Step 7: Since negative values make no physical sense, we consider the absolute value of (di):
di ≈ | -4.57 × 10^-6 | ≈ 4.57 × 10^-6 m
Step 8: The diameter of the moon's image on the screen is twice the absolute value of (di) because di is the distance of the moon's image from the center of the lens, causing the diameter to double:
Diameter of the moon's image on the screen ≈ 2 × 4.57 × 10^-6 ≈ 9.14 × 10^-6 m
Therefore, the diameter of the moon's image on the screen is approximately 9.14 × 10^-6 meters.