In separate experiments, an object is placed 12.0 cm in front of two different mirrors: a
flat mirror and a convex mirror. The image formed by the convex mirror is 4.0 cm closer
to the mirror than the image formed by the plane mirror. Determine the focal length of
the convex mirror
so the image distance for the flat mirror is 12cm behind the mirror. That locates image distance for the convex mirror. Use the lens formula, watch signs on f.
Well, well, well! We have some mirror mischief going on here! Let's get to the bottom of it, shall we?
We know that the image formed by the convex mirror is 4.0 cm closer to the mirror than the image formed by the plane mirror. So, if we consider the distance between the object and the plane mirror as x, the distance between the object and the convex mirror is x - 4.0 cm.
Now, let's use the mirror formula to solve this mystery! The mirror formula states that 1/focal length = 1/object distance + 1/image distance.
For the plane mirror, we have:
1/f_plane = 1/x + 1/image_distance_plane
And for the convex mirror, we have:
1/f_convex = 1/(x - 4.0 cm) + 1/image_distance_convex
Since the images formed by both mirrors are virtual, the image distances will be negative. A negative image distance means that the image is formed on the same side as the object.
Now, let's subtract the two equations to eliminate the image distances:
1/f_plane - 1/f_convex = (1/x + 1/image_distance_plane) - (1/(x - 4.0 cm) + 1/image_distance_convex)
Simplifying the equation, we get:
1/f_plane - 1/f_convex = (1/x - 1/(x - 4.0 cm))
Now, let's solve for the focal length of the convex mirror, which is what we're after:
1/f_convex = (1/x - 1/(x - 4.0 cm)) + 1/f_plane
By rearranging the equation, we can isolate the focal length:
f_convex = 1/[(1/x - 1/(x - 4.0 cm)) + 1/f_plane]
Now, all we need is the value of x (the object distance from the plane mirror) and the focal length of the plane mirror. Plug those values into the equation, and voila! You've uncovered the focal length of the convex mirror.
Remember, my friend, patience is the key when dealing with mirrors and their reflections. Happy solving!
To determine the focal length of the convex mirror, we can use the mirror formula:
1/f = 1/di + 1/do
where:
f = focal length
di = image distance
do = object distance
In this case, we know that the object distance for both mirrors is 12.0 cm and the difference in image distances is 4.0 cm.
Let's assume the image distance for the plane mirror is di1 and for the convex mirror is di2.
Using the given data, we have:
di1 - do = 4.0 cm
Since the object distance for both mirrors is the same, we can rearrange the equation:
di1 = di2 + 4.0 cm
Substituting this value into the mirror formula, we have:
1/f = 1/di2 + 1/(di2 + 4.0 cm)
Next, we can substitute the value of the object distance (do = 12.0 cm) into the equation:
1/f = 1/di2 + 1/(di2 + 4.0 cm)
1/f = (di2 + 4.0 cm + di2) / (di2 * (di2 + 4.0 cm))
Simplifying further:
1/f = (2di2 + 4.0 cm) / (di2^2 + 4.0 cm * di2)
We can solve for the value of di2 by substituting it into the equation:
di2^2 + 4.0 cm * di2 - (2di2 + 4.0 cm) * f = 0
This is a quadratic equation in di2. By solving for di2, we can find the focal length of the convex mirror.
To determine the focal length of the convex mirror, we can use the mirror formula:
1/f = 1/di + 1/do
Where:
f = focal length of the mirror
di = image distance
do = object distance
Given that the object is placed 12.0 cm in front of both mirrors, we have:
For the flat mirror:
do = 12.0 cm
di = ?
Since the flat mirror creates a virtual image at the same distance behind the mirror as the object is in front, the image distance would also be -12.0 cm.
For the convex mirror:
do = 12.0 cm
di = -12.0 cm - 4.0 cm = -16.0 cm (4.0 cm closer to the mirror)
Plugging in the values into the mirror formula:
1/f = 1/di + 1/do
1/f = 1/-16.0 cm + 1/12.0 cm
To simplify the equation, we can find the common denominator for the fractions:
1/f = (12.0 cm - 16.0 cm)/(-16.0 cm * 12.0 cm)
1/f = -4.0 cm / -192.0 cm^2
Simplifying further:
1/f = 1/48.0 cm
Therefore, the focal length of the convex mirror is 48.0 cm.