A candle is placed 17.0 cm in front of a convex mirror. When the convex mirror is replaced with a plane mirror, the image moves 6.0 cm farther away from the mirror. Find the focal length of the convex mirror.
_______cm
Well then, the image distance is 11cm in back of the mirror. Use the lens equation. Watch signs.
-12.36
To find the focal length of the convex mirror, we can use the mirror equation:
1/f = 1/di + 1/do
Where:
- f is the focal length of the mirror
- di is the image distance (distance between the mirror and the image)
- do is the object distance (distance between the mirror and the object)
Given:
- The object distance, do = 17.0 cm
- The change in the image distance, Δdi = 6.0 cm (since the image moves 6.0 cm farther away)
Using the mirror equation, we can rearrange it to solve for the focal length (f):
1/f = 1/di + 1/do
Substituting the values:
1/f = 1/(do + Δdi) + 1/do
Now, we can substitute the given values:
1/f = 1/(17.0 + 6.0) + 1/17.0
Simplifying this equation, we get:
1/f = 1/23.0 + 1/17.0
To add these fractions, we need to find a common denominator, which is equal to the product of the two denominators:
1/f = (17.0 + 23.0) / (17.0 * 23.0) + (23.0 + 17.0) / (17.0 * 23.0)
1/f = 40 / (17.0 * 23.0) + 40 / (17.0 * 23.0)
1/f = (40 + 40) / (17.0 * 23.0)
1/f = 80 / (17.0 * 23.0)
Now, we can simplify further:
1/f ≈ 80 / 391
To find f, we take the reciprocal of both sides of the equation:
f ≈ 391 / 80
Calculating this value using a calculator:
f ≈ 4.89 cm
Therefore, the focal length of the convex mirror is approximately 4.89 cm.
To find the focal length of the convex mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f is the focal length
v is the image distance
u is the object distance
We are given that the candle is placed 17.0 cm in front of the convex mirror, so the object distance is u = -17.0 cm (negative because the object is in front of the mirror).
When the convex mirror is replaced with a plane mirror, the image moves 6.0 cm farther away from the mirror. This means that the image distance when the mirror is replaced with a plane mirror is v + 6.0 cm.
Using the mirror formula, we can now substitute the values:
1/f = 1/(v + 6) - 1/(-17)
Simplifying this equation:
1/f = -1/17 - 1/(v + 6)
To get rid of the fractions, we can take the reciprocals:
1/f = -(v + 6)/(17(v + 6)) - 17/(17(v + 6))
Combining the terms:
1/f = (-v - 6 - 17)/(17(v + 6))
Further simplifying:
1/f = (-v - 23)/(17(v + 6))
To isolate f, we can take the reciprocal of both sides:
f = 17(v + 6)/(-v - 23)
So, the focal length of the convex mirror is:
f = 17(v + 6)/(-v - 23) cm