An object with a height of 12cm is placed 4cm in front of a concave mirror whose radius of curvature is 16cm.
To find the image formed by the concave mirror, we can use the mirror equation:
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).
First, let's find the focal length of the concave mirror using the formula:
f = R/2
where R is the radius of curvature.
Given that the radius of curvature is 16 cm, we have:
f = 16/2 = 8 cm
Now, let's find the object distance (u). The object distance is the distance of the object from the mirror, which is given as 4 cm.
u = -4 cm (since the object is placed in front of the mirror, the distance is negative)
Finally, we can substitute the values of f and u into the mirror equation to find the image distance (v):
1/8 = 1/v - 1/-4
To simplify the equation, we can multiply through by 8v(-4) to eliminate the denominators:
-4v + 32 = -32v
Now, let's solve for v:
32v - 4v = 32
28v = 32
v = 32/28
v = 1.14 cm (rounded to two decimal places)
Therefore, the image formed by the concave mirror is located 1.14 cm in front of the mirror.
To determine the image of the object produced by the concave mirror, you 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), and
- u is the object distance (distance of the object from the mirror).
In this case, we are given the object height (h) = 12 cm, the object distance (u) = -4 cm (because it is placed in front of the mirror), and the radius of curvature (R) = -16 cm (negative since it is a concave mirror).
The focal length (f) of a concave mirror is half the radius of curvature (f = R/2).
Let's calculate the focal length first:
f = -16 cm / 2 = -8 cm
Now we can substitute the given values into the mirror formula and solve for v:
1/(-8 cm) = 1/v - 1/(-4 cm)
To simplify, let's combine the fractions:
-1/8 = 1/v + 1/4
To add fractions with different denominators, we find the common denominator, which is 8v:
-1/8 = (2 + v) / (8v)
To eliminate the fraction, we can cross-multiply:
-8v = 2 + v
Now let's solve for v:
-8v - v = 2
-9v = 2
v = -2/9 cm
Since the negative sign indicates that the image is formed on the same side as the object (i.e., the left side), and the magnitude of the image distance is positive, the image is virtual and upright.
Finally, to find the magnification (M), we can use the formula:
M = -v/u
Substituting the values:
M = -(-2/9 cm) / (-4 cm) = 1/2
Therefore, the image formed by the concave mirror is virtual, upright, and its height is half the height of the object.