An object is placed in front of a convex mirror with a 22.6-cm radius of curvature. A virtual image half the size of the object is formed. At what distance is the object from the mirror? Take a positive answer to be in front of the mirror, and a negative answer to be behind the mirror.
To solve this problem, we can use the mirror equation:
1/f = 1/do + 1/di
Where:
- f is the focal length of the convex mirror
- do is the object distance
- di is the image distance
The focal length (f) of a convex mirror is always half the radius of curvature (R):
f = R/2
In this case, the radius of curvature (R) is given as 22.6 cm, so we can calculate the focal length:
f = 22.6 cm / 2 = 11.3 cm
Next, we can use the given information that the virtual image formed is half the size of the object. In terms of magnification (m), we know that the magnification is given by:
m = -di/do
Since the virtual image is half the size of the object, the magnification (m) is -0.5:
-0.5 = -di/do
Therefore, di = 0.5 * do
Now, we can substitute the values for f and di into the mirror equation:
1/11.3 = 1/do + 1/(0.5 * do)
Now, we can solve for do:
1/11.3 = (1 + 2)/do
1/11.3 = 3/do
Cross-multiplying:
do = 11.3 * 3
do = 33.9 cm
Since a positive answer represents an object in front of the mirror, the object is located 33.9 cm in front of the convex mirror.