When a converging lens is used in a camera (as in the first drawing), the film must be placed at a distance of 0.240 m from the lens to record an image of an object that is 4.30 m from the lens. The same lens is then used in a projector (as in the second drawing), with the screen 0.560 m from the lens. How far from the projector lens should the film be placed?
The film should be placed at a distance of 2.25 m from the projector lens.
To find the distance at which the film should be placed in the projector, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the distance of the image from the lens (screen in this case)
- u is the distance of the object from the lens (film in this case)
First, let's calculate the focal length of the lens:
Given: v1 = 0.240 m, u1 = 4.30 m
1/f = 1/0.240 - 1/4.30
1/f = 4.1667 - 0.2326
1/f = 3.9341
f ≈ 0.254 m
Now, let's calculate the distance of the film from the lens:
Given: v2 = 0.560 m
Using the lens formula again:
1/f = 1/0.560 - 1/u2
Rearranging the formula, we get:
1/u2 = 1/f - 1/v2
1/u2 = 1/0.254 - 1/0.560
1/u2 = 3.9370 - 1.7857
1/u2 = 2.1513
u2 ≈ 0.464 m
Therefore, the film should be placed at a distance of approximately 0.464 m from the projector lens.
To determine the distance from the projector lens to the film, we can use the lens formula:
1/f = 1/d₀ + 1/d_i
Where:
- f is the focal length of the lens
- d₀ is the distance of the object from the lens
- d_i is the distance of the image from the lens
Given:
- d₀ = 4.30 m (distance of the object from the lens)
- d_i = 0.560 m (distance of the image from the lens)
Since it is the same lens being used, the focal length (f) remains the same. So, we can use the values to solve for the new distance (d₀) in the projector scenario.
Let's calculate it step-by-step:
Step 1: Rewrite the lens formula
1/f = 1/d₀ + 1/d_i
Step 2: Substitute the known values
1/f = 1/4.30 + 1/0.560
Step 3: Convert the fractions to a common denominator
1/f = (0.560 + 4.30)/(4.30 * 0.560)
Step 4: Simplify the fraction
1/f = 4.860/2.408
Step 5: Invert the fraction
f = 2.408/4.860
Step 6: Calculate
f ≈ 0.496 m
Now that we have the focal length of the lens (f) in the projector scenario, we can determine the distance (d₀) from the projector lens to the film using the same lens formula.
1/f = 1/d₀ + 1/d_i
Step 7: Substitute the known values
1/0.496 = 1/d₀ + 1/0.560
Step 8: Convert the fractions to a common denominator
1/0.496 = (0.560 + d₀)/(0.560 * d₀)
Step 9: Simplify the fraction
1/0.496 = (0.560 + d₀)/(0.560d₀)
Step 10: Cross-multiply
0.496(0.560d₀) = (0.560 + d₀)
Step 11: Distribute and simplify
0.2776d₀ = 0.560 + d₀
Step 12: Move d₀ terms to one side
0.2776d₀ - d₀ = 0.560
-0.7224d₀ = 0.560
Step 13: Divide both sides by -0.7224
d₀ ≈ -0.772 m
Note: The negative value indicates that the film needs to be placed on the opposite side of the lens. Since distances cannot be negative, it means that the film should be placed on the opposite side at a distance of 0.772 m from the projector lens.