An object is placed in front of a convex mirror with radius of curvature = 17 cm. The object's height is 3.5 cm and it is initially 25 cm in front of the mirror. Find the (a) location and (b) height of the image. Next the object moves closer to the mirror, so the object distance is 5.1 cm. Again, find the (c) location and (d) height of the image. (e) What is the ratio of the image height when the object distance is 5.1 cm to that when the object distance is 25 cm?
To solve this problem, we'll use the mirror equation and the magnification formula for convex mirrors.
(a) Location of the image:
We'll use the mirror equation:
1/f = 1/do + 1/di
where f is the focal length of the mirror, do is the object distance, and di is the image distance.
Since the mirror is convex, the focal length (f) is positive, given by:
f = R/2
where R is the radius of curvature of the mirror.
Substituting the values R = 17 cm and do = 25 cm into the mirror equation, we have:
1/(17/2) = 1/25 + 1/di
Solving this equation, we can find the value of di, which represents the location of the image.
(b) Height of the image:
We'll use the magnification formula:
m = -di/do
where m is the magnification, di is the image distance, and do is the object distance.
Substituting the values do = 25 cm and di (from part a) into the magnification formula, we can calculate the value of m.
Next, let's move on to the second scenario:
(c) Location of the image:
Again, we'll use the mirror equation with the new value of do = 5.1 cm:
1/(17/2) = 1/5.1 + 1/di
Solving this equation, we can find the value of di, representing the location of the image.
(d) Height of the image:
Using the magnification formula, m = -di/do, with the new values do = 5.1 cm and di (from part c), we can calculate the value of m.
(e) Ratio of image height:
The ratio of image heights for the two scenarios can be calculated by dividing the height of the image (from part d) when do = 5.1 cm by the height of the image when do = 25 cm.
By following these steps, you should be able to find the location and height of the image for each scenario, as well as the ratio of the image heights.