A dime .334 m away from, and on the optical axis of, a concave spherical mirror produces an image on the same side of the mirror as the source. The image is .093 m away from the mirror. If the dime is moved on the axis to .229 m from the mirror, how far away from the mirror is the image now?
What is the radius of the sphere of which the mirror is a section?
You don't need the radius of the mirror in this case. Since it and the focal length stay the same,
1/.334 + 1/.093 = 1/.229 + 1/x
Solve for the 'x', which is the new image distance from the mirror.
1/x = 2.994 + 10.753 - 4.367 = 9.38
x = 0.107 m
To find the radius of the spherical mirror, we can use the mirror equation:
1/f = 1/do + 1/di
Where:
- f is the focal length of the mirror,
- do is the object distance (distance of the object from the mirror),
- and di is the image distance (distance of the image from the mirror).
In the given problem, we are initially given that the object distance (do) is 0.334 m, the image distance (di) is 0.093 m, and the image is formed on the same side as the object.
Using the mirror equation, we can find the focal length of the mirror:
1/f = 1/do + 1/di
1/f = 1/0.334 + 1/0.093
Now, let's calculate:
1/f = 2.994 + 10.753
1/f = 13.747
Therefore, the focal length (f) of the mirror is approximately 0.0727 m.
Now, to find the radius of the mirror, we can use the formula:
f = R/2
Where:
- R is the radius of the mirror.
Substituting the value of f into the formula:
0.0727 = R/2
Now, let's solve for R:
R = 2 * 0.0727
R = 0.1454 m
Therefore, the radius of the sphere of which the mirror is a section is approximately 0.1454 m.