The lens and mirror in the figure below are separated by d = 1.00 m and have focal lengths of +74.8 cm and -42.2 cm, respectively. An object is placed p = 1.00 m to the left of the lens as shown.
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To determine the position and nature of the final image formed by the lens and mirror system, we can use the lens formula and mirror formula.
The lens formula is given by:
1/f = 1/v - 1/u
where f is the focal length of the lens, v is the image distance, and u is the object distance.
Similarly, the mirror formula is given by:
1/f = 1/v' + 1/u'
where f is the focal length of the mirror, v' is the image distance formed by the mirror, and u' is the object distance for the mirror.
Since the object is placed 1.00 m to the left of the lens, the object distance u for the lens is -1.00 m.
Now we can use the lens formula to find the image distance v for the lens.
1/f = 1/v - 1/u
1/0.748 m^-1 = 1/v - 1/-1.00 m^-1
Converting the negative sign, we get:
1/0.748 m^-1 = 1/v + 1.00 m^-1
Rearranging the equation, we get:
1/v = 1/0.748 m^-1 - 1.00 m^-1
1/v = 1.34 m^-1 - 1.00 m^-1
1/v = 0.34 m^-1
Taking the reciprocal of both sides, we get:
v = 1 / 0.34 m^-1
v ≈ 2.94 m
So, the image distance for the lens is approximately 2.94 m.
Now, using the mirror formula, we can find the image distance v' for the mirror.
1/f = 1/v' + 1/u'
1/-0.422 m^-1 = 1/v' + 1/2.94 m^-1
Rearranging the equation, we get:
1/v' = 1/-0.422 m^-1 - 1/2.94 m^-1
1/v' = -2.37 m^-1 - 0.34 m^-1
1/v' ≈ -2.71 m^-1
Taking the reciprocal of both sides, we get:
v' ≈ -1 / 2.71 m^-1
v' ≈ -0.36 m
So, the image distance for the mirror is approximately -0.36 m.
The negative sign indicates that the final image formed by the mirror is virtual, meaning it is formed on the same side as the object.
Therefore, the final image formed by the lens and mirror system is located approximately 0.36 m to the left of the mirror and is a virtual image.