A 4.5 cm tall object is placed 20 cm in front of a spherical mirror. It is desired to produce a virtual image that is upright and 3.5 cm tall.
(a) What type of mirror should be used?
concave mirror
convex mirror
(b) Where is the image located? (Your answer should be positive if the image is in front of the mirror and negative if the image is behind it.)
cm
(c) What is the focal length of the mirror?
cm
(d) What is the radius of curvature of the mirror?
(a) The mirror that should be used is a concave mirror.
(b) The image is located in front of the mirror because it is a virtual image. Therefore, the answer is positive.
(c) To determine the focal length of the mirror, we can use the mirror equation:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance (positive for virtual image)
u = object distance (negative for object in front of the mirror)
Given:
u = -20 cm (object distance)
v = image distance (unknown)
We can rearrange the equation as follows:
1/f = 1/v + 1/(-20 cm)
Now, substitute the values and solve for f:
1/f = 1/v - 1/20 cm
1/v = 1/f + 1/20 cm
v = 20 cm / (1/f + 1/20 cm)
Since we want to produce a virtual image, the image distance (v) must be positive. Therefore, we need a positive value for v.
To find the focal length (f), we can choose an arbitrary value for v, such as 30 cm. Plug this value into the equation:
v = 20 cm / (1/f + 1/20 cm)
30 cm = 20 cm / (1/f + 1/20 cm)
Solve for f:
1/f + 1/20 cm = 20 cm / 30 cm
1/f = 20 cm / 30 cm - 1/20 cm
1/f = (20 - 3) / 60 cm
1/f = 17 / 60 cm
f = 60 cm / 17
f ≈ 3.53 cm
Therefore, the focal length of the mirror is approximately 3.53 cm.
(d) The radius of curvature (R) of a spherical mirror is simply twice the focal length:
R = 2f
R ≈ 2(3.53 cm)
R ≈ 7.06 cm
Therefore, the radius of curvature of the mirror is approximately 7.06 cm.
To solve this problem, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance from the mirror
u = object distance from the mirror
(a) To determine the type of mirror (convex or concave), we need to analyze the characteristics of the image formed. In this case, the object height is positive and the image height is also positive, which means that the image is upright. This implies that a concave mirror is used because a convex mirror always produces an inverted image.
Therefore, the correct answer is a concave mirror.
(b) Next, we need to determine the image distance, v. The given object distance, u, is 20 cm, and we need to produce a virtual image that is upright and 3.5 cm tall. Since the image is virtual, it will be on the same side of the mirror as the object, so the image distance, v, will be positive.
(c) To find the focal length, f, we can rearrange the mirror formula to solve for f:
1/f = 1/v - 1/u
Substituting the known values:
1/f = 1/v - 1/20
Now, we can substitute the image distance, v, to solve for f.
(d) To find the radius of curvature, R, we can use the relationship between the focal length and the radius of curvature for a spherical mirror:
R = 2f
Substituting the obtained focal length, f, we can find the radius of curvature, R.
Let's proceed with finding the values for (b), (c), and (d) by substituting the given information into the equations.
Part (a) should be convex.
Part (b)You have to put the height of your image over the height of your object, like a ratio. Once you do that, multiply that answer by the "20cm." Your answer should be a negative.
Part (c) Use the equation: 1/do+1/di=1/f
Part (d) Multiply the answer you got for part (c) by two