The lens and mirror in the figure below are separated by d = 1.00 m and have focal lengths of +74.9 cm and -56.4 cm, respectively. An object is placed p = 1.00 m to the left of the lens as shown.
To solve this problem, we can use the lens formula and mirror formula. The lens formula relates the object distance (p), image distance (q), and focal length (f) of a lens, while the mirror formula relates the object distance (p), image distance (q), and focal length (f) of a mirror.
The lens formula is given by:
1/f = 1/p + 1/q
The mirror formula is given by:
1/f = 1/p + 1/q
Given:
Focal length of the lens (f_l) = +74.9 cm = +0.749 m (positive sign indicates a converging lens)
Focal length of the mirror (f_m) = -56.4 cm = -0.564 m (negative sign indicates a concave mirror)
Object distance (p) = 1.00 m
We need to find the image distance (q) for the lens and mirror system.
Step 1: Calculate the image distance (q) for the lens using the lens formula.
1/f_l = 1/p + 1/q_l
Substituting the values into the formula:
1/0.749 = 1/1 + 1/q_l
Simplifying:
1/0.749 = 1 + 1/q_l
1/q_l = 1/0.749 - 1
1/q_l = (1 - 0.749) / 0.749
1/q_l = 0.251 / 0.749
1/q_l = 0.3358
Taking the reciprocal of both sides:
q_l = 1 / 0.3358
q_l ≈ 2.979 m
So, the image distance (q_l) for the lens is approximately 2.979 meters.
Step 2: Calculate the image distance (q_m) for the mirror using the mirror formula.
1/f_m = 1/p + 1/q_m
Substituting the values into the formula:
1/-0.564 = 1/1 + 1/q_m
Simplifying:
-1.7702 = 1 + 1/q_m
1/q_m = -1 -(-1.7702)
1/q_m = 1.7702 - 1
1/q_m = 0.7702
Taking the reciprocal of both sides:
q_m = 1 / 0.7702
q_m ≈ 1.298 m
So, the image distance (q_m) for the mirror is approximately 1.298 meters.
Therefore, the object placed 1.00 m to the left of the lens will form an image at a distance of approximately 2.979 m to the right of the lens and at a distance of approximately 1.298 m to the left of the mirror.