A 3.0 cmm high object is placed 10 cm from a concave mirror whose radius of curvature is 10 cm. Where is the image located?
To calculate the location of the image formed by the concave mirror, we can use the mirror equation:
1/f = 1/o + 1/i,
where "f" is the focal length of the mirror, "o" is the object distance, and "i" is the image distance.
Given:
Object height (h) = 3.0 cm,
Object distance (o) = -10 cm (since the object is placed in front of the mirror),
Radius of curvature (R) = -10 cm (note the negative sign indicates a concave mirror).
Now, let's determine the focal length (f) using the formula:
f = R/2.
f = -10 cm / 2 = -5 cm.
Substituting the given values into the mirror equation:
1/f = 1/o + 1/i,
1/(-5 cm) = 1/(-10 cm) + 1/i,
Simplifying the equation, we get:
-1/5 = 1/(-10) + 1/i,
Now, let's solve for 1/i:
-1/5 - 1/(-10) = 1/i,
Multiplying through by (-10) to eliminate the fractions:
-2 + 1/(-10) = 1/i,
Simplifying further:
-2 - 1/10 = 1/i,
-20/10 - 1/10 = 1/i,
-21/10 = 1/i.
Invert both sides of the equation to solve for "i":
10/-21 = i,
Therefore, the image distance "i" is approximately -0.476 cm.
The negative sign of the image distance indicates that the image formed is virtual, upright, and located behind the mirror. The absolute value of the image distance (0.476 cm) represents the distance behind the mirror where the image is formed.