To determine the position of the object in front of a concave mirror, we can use the mirror formula:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
where:
- [tex]\( f \)[/tex] is the focal length of the mirror,
- [tex]\( v \)[/tex] is the image distance,
- [tex]\( u \)[/tex] is the object distance.
Since we have a concave mirror, both the focal length and image distance are taken as negative values. However, provided values are:
- Image distance, [tex]\( v = 20.0 \)[/tex] cm (in front of the mirror, hence [tex]\( v \)[/tex] should be treated as negative [tex]\( v = -20.0 \text{ cm} \)[/tex]).
- Focal length, [tex]\( f = 12.0 \)[/tex] cm (concave mirror, hence [tex]\( f \)[/tex] should be treated as negative [tex]\( f = -12.0 \text{ cm} \)[/tex]).
We need to find the object distance, [tex]\( u \)[/tex].
Using the mirror formula:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
Substitute the given values into the formula:
[tex]\[
\frac{1}{-12.0} = \frac{1}{-20.0} + \frac{1}{u}
\][/tex]
First, calculate [tex]\(\frac{1}{-20.0}\)[/tex]:
[tex]\[
\frac{1}{-20.0} = -0.05
\][/tex]
Now substitute back into the formula and solve for [tex]\(\frac{1}{u}\)[/tex]:
[tex]\[
\frac{1}{-12.0} = -0.05 + \frac{1}{u}
\][/tex]
Calculate [tex]\(\frac{1}{-12.0}\)[/tex]:
[tex]\[
\frac{1}{-12.0} = -0.0833
\][/tex]
Now rearrange the equation:
[tex]\[
-0.0833 = -0.05 + \frac{1}{u}
\][/tex]
Subtract [tex]\(-0.05\)[/tex] from both sides:
[tex]\[
-0.0833 + 0.05 = \frac{1}{u}
\][/tex]
[tex]\[
-0.0333 = \frac{1}{u}
\][/tex]
To find [tex]\(u\)[/tex], we take the reciprocal of the result:
[tex]\[
u = \frac{1}{-0.0333} = -30.0 \text{ cm}
\][/tex]
The negative sign indicates that the object is located in front of the mirror. So the object distance, [tex]\(u\)[/tex], is:
[tex]\[
30.0 \text{ cm}
\][/tex]
Thus, the object is located 30.0 cm in front of the mirror.
