A coin 1.9 cm in diameter is held 40 cm from a concave spherical mirror of 30 cm radius of curvature. Locate the image of the coin
Then what is the diameter of coin?
To locate the image of the coin, we can use the mirror equation:
1/f = 1/di + 1/do
where f is the focal length of the mirror, di is the distance of the image from the mirror, and do is the distance of the object from the mirror.
In this case, the mirror is a concave spherical mirror with a radius of curvature of 30 cm, so the focal length (f) is half the radius of curvature:
f = 30 cm / 2 = 15 cm
The object distance (do) is the distance from the mirror to the coin, which is given as 40 cm.
Using the mirror equation:
1/15 = 1/di + 1/40
Rearranging the equation to solve for the image distance (di):
1/di = 1/15 - 1/40
= (40 - 15) / (15 * 40)
= 25 / 600
= 1 / (600/25)
= 1/24
di = 24 cm
Therefore, the image of the coin is located 24 cm from the concave spherical mirror.
Now, to find the diameter of the coin, let's consider the magnification (m):
m = -di / do
where di is the image distance and do is the object distance. The negative sign represents an inverted image.
Substituting the values:
m = -24 cm / 40 cm
= -0.6
The magnification tells us the ratio of the image height to the object height. Since the coin is coin is located beyond the focal point (di > f), it is a reduced image.
Now, let's use the magnification formula:
m = hi / ho = -di / do
where hi is the image height and ho is the object height.
Since the object (coin) is a circular disk, the height is equal to the diameter. Let's consider the diameter of the object coin as 'do':
-0.6 = di / do
Rearranging the equation for the image height (hi):
hi = -0.6 * do
Substituting the object height:
hi = -0.6 * 1.9 cm
= -1.14 cm
Since magnification is the ratio of the image height to the object height, we can say:
hi / ho = -0.6
Substituting the equation for the image and object heights:
-1.14 cm / do = -0.6
Rearranging the equation for the diameter of the coin (do):
do = -1.14 cm / -0.6
= 1.9 cm / 0.6
= 3.167 cm
Therefore, the diameter of the coin is approximately 3.167 cm.