A certain LCD projector contains a single thin lens. An object 23.5 mm high is to be projected so that its image fills a screen 1.87 m high. The object-to-screen distance is 3.05 m.
(a) Determine the focal length of the projection lens.
mm
(b) How far from the object should the lens of the projector be placed in order to form the image on the screen?
mm
To determine the focal length of the projection lens, you can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens.
- v is the image distance (distance of the image from the lens).
- u is the object distance (distance of the object from the lens).
In this case, u = 3.05 m and the height of the object, h, is known as 23.5 mm.
First, let's convert the object height to meters:
h = 23.5 mm = 0.0235 m.
Now, we can find the image distance:
v = 1.87 m.
Now, substitute the values into the lens formula:
1/f = 1/v - 1/u
1/f = 1/1.87 - 1/3.05
Solve for 1/f:
1/f = (3.05 - 1.87)/(1.87 * 3.05)
Now, calculate 1/f:
1/f = 0.623/5.723
Finally, find f by taking the reciprocal of 1/f:
f = 1/(0.623/5.723)
f ≈ 9.191 mm
So, the focal length of the projection lens is approximately 9.191 mm.
To determine the distance from the object to the lens, we can use the lens formula again:
1/f = 1/v - 1/u
Now, we know the focal length f and the image distance v, which is 1.87 m.
Substitute the values into the lens formula:
1/f = 1/1.87 - 1/u
Since we want to find the distance from the object to the lens (u), we rearrange the formula:
1/u = 1/f - 1/v
Substitute the known values:
1/u = 1/9.191 - 1/1.87
Simplify:
1/u = (1.087 - 0.534)/9.191
Now, calculate 1/u:
1/u = 0.553/9.191
Finally, find u by taking the reciprocal of 1/u:
u = 1/(0.553/9.191)
u ≈ 16.628 mm
Therefore, the lens of the projector should be placed approximately 16.628 mm from the object in order to form the image on the screen.