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Chapter 4: Problem 4

The full Moon subtends an angle of approximately 30 arc min. How large would the image of the Moon be on your film if you used a \(500 \mathrm{mm}\) focal length lens for your camera?

Short Answer

Expert verified
The size of the Moon's image on film would be approximately 4.36 mm.

Step by step solution

01

Understand the Problem

The task is to find the size of the Moon's image on film, given the angle it subtends and the focal length of the lens.
02

Convert Arc Minutes to Radians

The angle subtended by the Moon is 30 arc minutes. There are 60 arc minutes in a degree.First, convert 30 arc minutes to degrees: \[30 \text{ arc min} \times \frac{1 \text{ degree}}{60 \text{ arc min}} = 0.5 \text{ degrees}\]\Now, convert degrees to radians using the conversion factor \(\frac{\text{π radians}}{180 \text{ degrees}}\): \[0.5 \text{ degrees} \times \frac{\text{π radians}}{180 \text{ degrees}} = \frac{0.5 \text{π}}{180} = \frac{π}{360} \text{ radians}\]
03

Use Lens Formula

For small angles, the size of the image can be found using the simple lens formula: \[ \theta = \frac{h}{f} \] where \( \theta \) is the angle in radians, \( h \) is the height of the image, and \( f \) is the focal length. Rearrange this to solve for \( h \): \[ h = \theta f\]\
04

Calculate the Size of the Image

Substitute \(\theta = \frac{\text{π}}{360} \text{ radians}\) and \(f = 500 \text{ mm}\) into the rearranged formula: \[ h = \left( \frac{\text{π}}{360} \right) \times 500 \text{ mm} = \frac{500 \text{ π}}{360} \text{ mm} = 4.36 \text{ mm}\] Therefore, the size of the Moon's image is approximately 4.36 mm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angle subtended
In astronomy, the concept of 'angle subtended' helps us understand how large an object appears in the sky. The size of an object (like the Moon) is measured by the angle it forms with the observer's eye. This is called the angle subtended. For instance, the full Moon subtends an angle of about 30 arc minutes. A larger subtended angle means the object appears bigger, and a smaller angle means it appears smaller. This angle does not represent the actual size of the object but how large it appears to us from Earth.
arc minutes to radians conversion
Astronomers often need to convert the angle measurements from arc minutes to radians because radians are a more convenient unit in calculations. One degree has 60 arc minutes and one radian equals about 57.3 degrees. To convert arc minutes to radians:
  • First, convert arc minutes to degrees.
    For example, 30 arc minutes is 0.5 degrees (30/60).
  • Next, multiply the degrees by π/180 to get radians.
  • Example: 0.5 degrees × π/180 = π/360 radians.
This process simplifies many calculations in optics and astronomy.
lens formula in optics
The lens formula in optics relates the angle, the height of the image, and the focal length of the lens. For small angles, the formula is:
\[ \theta = \frac{h}{f} \]
Where θ is the angle in radians, h is the image size, and f is the focal length of the lens. By rearranging the formula, we can solve for the image size (h) as follows:
\[ h = \theta f \]
This means that if we know the angle θ and the focal length f of the lens, we can easily calculate the height of the image h.
image size calculation
Finally, calculating the size of the image involves combining the angle in radians and the lens formula. Here’s how:
  • Convert the angle to radians: 30 arc minutes = π/360 radians.
  • Use the lens formula in optics:
    \( h = \theta f \)
    Insert θ (π/360 radians) and f (500 mm).
  • Calculate the image size:
    \( h = \frac{\pi}{360} \times 500 \text{ mm} \approx 4.36 \text{ mm} \)
The Moon's image size on the film will be approximately 4.36 mm, given a 500 mm focal length lens.

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