An object is placed in front of a concave mirror, 16.0 cm from the mirror's focal point. The image formed by the mirror is two times farther away from the focal point. Calculate the focal length of the mirror.
There are actually two possible image distances di that satisfy the statement of the problem. Enter the larger of the two image distances.
Enter the smaller of the two image distances.
To answer this problem since it provided the magnification and do I used the formula: m= -di/do to get the value of di
so I said 2= -di/16 therefore I said di=-32 I figured that would be the smaller value and the larger value would be +32 but that was marked wrong.
Also to get my focal point I used the formula: 1/f= 1/di + 1/do and I also got 32 as the answer but that was also marked wrong. What am I doing wrong?
To solve this problem, let's first clarify the variables we have:
- di: image distance
- do: object distance
- f: focal length
- m: magnification
You correctly used the magnification formula: m = -di/do, and from the given information, we know that m = 2. So, we have:
2 = -di/16
You made a mistake in your calculation. Let's correct it:
-2 = di/16
By cross-multiplying, we find:
di = -32 cm
Now, this negative sign indicates that the image formed is virtual, which means it is on the same side of the mirror as the object. Therefore, the image distance is -32 cm.
Next, to calculate the focal length, we can use the lens formula: 1/f = 1/di + 1/do.
Plugging in the values, we get:
1/f = 1/-32 + 1/16
When you simplify this equation, you should get:
1/f = -1/32 + 1/16
Common denominator for the right side is 32, so:
1/f = -2/32 + 2/32
Combining the terms, we have:
1/f = 0/32
Since the right hand side is zero, we get:
1/f = 0
This indicates that the focal length is infinite.
Thus, there is no specific focal length for this concave mirror configuration. The mirror is diverging and does not have a finite focal length.
To find the focal length of the concave mirror, you can use the mirror equation:
1/f = 1/di + 1/do
where f is the focal length, di is the image distance, and do is the object distance.
In this problem, you are given that the object distance (do) is 16.0 cm. You need to find the image distance (di).
You correctly used the magnification formula m = -di/do to find di. Since the magnification is given as 2 (two times farther away from the focal point), you correctly set up the equation as:
2 = -di/16
To solve for di, you can multiply both sides of the equation by 16:
2 * 16 = -di
32 = -di
Now, you made a mistake in assuming that the image distance should be negative. For concave mirrors, the image distance is negative if the image is virtual (formed on the same side as the object). However, in this problem, it is stated that the image is two times farther away from the focal point, which means it is a real image (formed on the opposite side of the object). Therefore, the image distance should be positive.
So, di = 32 is the correct value.
Now, let's substitute the values into the mirror equation to find the focal length (f):
1/f = 1/di + 1/do
1/f = 1/32 + 1/16
1/f = (1 + 2)/32
1/f = 3/32
Now, we can solve for f by taking the reciprocal of both sides:
f = 32/3
So, the focal length of the concave mirror is approximately 10.67 cm.
To find the smaller and larger image distances, we need to consider that the mirror equation can also be written as:
1/f = 1/di + 1/do
So, if we fix the values of do and f, and vary di, there will be two possible image distances that satisfy the equation. One will be larger than the other.
In this case, the larger image distance (di) is 32 cm, and the smaller image distance (di) is 16 cm.