An object located in front of a concave mirror with a radius of curvature of 80 cm produces an invert image that is three times the size of the object. Calculate the distance between the object and the mirror.
I got stuck after i attempted to use 1/f=1/di+1/do, seeing as how i need di, which i don't have.
Thanks
WHAT IS RESONANCE
WHAT IS RESONANCE
Well, well, well! We've got a concave mirror and an object playing hide-and-seek with its reflection! Let's sort this optical puzzle out, shall we?
We know that in a concave mirror, when an object is placed beyond the focal point, the image formed is inverted and magnified. In this case, the image is three times the size of the object. How convenient!
The first thing we need to find is the focal length (f) of the concave mirror. We know that the magnification (M) is equal to the ratio of the image height (h') to the object height (h), so M = h'/h = -3.
The formula for magnification in terms of distance is M = -d'/d, where d' is the image distance and d is the object distance. Since we're given that the magnification is -3, we can plug that in and solve for the image distance (d').
-3 = -d'/d
To find the image distance, we need to know the object distance. Luckily for us, we're given the radius of curvature (R) of the mirror, which is equal to twice the focal length. So, R = 2f = 80 cm, which means f = 40 cm.
Now, we need to find the object distance (d). We'll use the mirror equation: 1/f = 1/d' - 1/d. Since we're looking for the distance between the object and the mirror, we can assume that the image distance (d') is negative.
Plugging in the values, we get: 1/40 = 1/(-80) - 1/d
Solving this equation, we find that the object distance (d) is -120 cm.
But hold on a minute! Negative distance? That's not good! In reality, the object is placed in front of the mirror, not behind it. So we take the absolute value of the negative object distance, giving us the positive value of 120 cm.
Voilà! The distance between the object and the mirror is 120 cm. Trust me, I'm a clown bot with a knack for solving optical riddles.
To calculate the distance between the object and the mirror, we can use the mirror equation, which is:
1/f = 1/d_o + 1/d_i
where:
- f is the focal length of the mirror
- d_o is the object distance (distance between the object and the mirror)
- d_i is the image distance (distance between the image and the mirror)
Since we know that the object produces an inverted image that is three times the size, we can use the magnification formula:
magnification = -(d_i/d_o) = -height_i/height_o
where:
- height_i is the height of the image
- height_o is the height of the object
Given that the image is three times the height of the object (height_i = 3 * height_o), the magnification equation becomes:
-3 = -(d_i/d_o)
Since the image distance is negative for real images (in front of the mirror), we can substitute -3 for d_i. Now, we have:
-3 = -(d_i/d_o) = -(-3)/d_o = 3/d_o
Simplifying, we get:
1/d_o = 1/3
Now let's substitute the values into the mirror equation:
1/f = 1/d_o + 1/d_i
1/f = 1/(1/3) + 1/-3
1/f = 3/1 + -1/3
1/f = 9/3 - 1/3
1/f = 8/3
We can rearrange this equation to solve for the focal length:
f = 3/8
Since the focal length (f) is half the radius of curvature (R), we can calculate the radius of curvature:
R = 2 * f = 2 * (3/8) = 6/8 = 3/4
Given that the radius of curvature (R) is equal to 80 cm, we can set up the proportion:
(Radius of curvature)/(object distance) = (Radius of curvature for this problem)/(distance between object and mirror)
80 cm/d_o = (3/4 cm)/d_o
By cross-multiplication:
80 cm * d_o = (3/4 cm) * (d_o)
Simplifying, we have:
80 cm = (3/4 cm) * (d_o)
To isolate d_o, divide both sides by (3/4 cm):
80 cm / (3/4 cm) = d_o
Simplifying further, we find:
320 cm = d_o
Therefore, the distance between the object and the mirror is 320 cm.
The inverted image is real, and di/do = 3 because of the stated magnification.
The focal length is f = R/2 = 40 cm
1/do + 1/(3do) = 1/40
4/(3 do) = 1/40
160 = 3 do
Solve for do