A light bulb is placed 19.6 cm in front of a concave mirror. When the concave mirror is replaced by a plane mirror in the same location, the bulb's image moves 5.50 cm closer to the mirror. Calculate the focal length of the concave mirror.
To solve this problem, we can use the mirror formula:
1/f = 1/v - 1/u
where:
- f is the focal length of the mirror
- v is the image distance (distance of the image from the mirror)
- u is the object distance (distance of the object from the mirror)
Let's assume that the object distance (u) for the concave mirror is 19.6 cm, and the image distance (v) is the distance at which the image moves when the concave mirror is replaced with a plane mirror, which is 5.50 cm closer to the mirror.
For the concave mirror:
1/f = 1/v - 1/u
Substituting the given values:
1/f = 1/(19.6 + 5.50) - 1/19.6
Simplifying the equation:
1/f = 1/25.1 - 1/19.6
Now, find the common denominator for the terms on the right side of the equation:
1/f = (19.6 - 25.1) / (19.6 * 25.1)
Calculating:
1/f = -5.5 / (19.6 * 25.1)
Now, taking the inverse on both sides of the equation:
f = (19.6 * 25.1) / -5.5
Calculating this equation, we find:
f ≈ -88.4727285 cm
Therefore, the focal length of the concave mirror is approximately -88.47 cm. Note that the negative sign indicates that it is a concave mirror.