A concave spherical mirror has a radius of curvature of 18 cm. Locate the images for object distances as given below. In each case, state whether the image is real or virtual and upright or inverted, and find the magnification. (If an answer does not exist, enter DNE. If an answer is infinity, enter INFINITY.)

(a)
p = 9 cm

image distance cm
image orientation
magnification

(b)
p = 18 cm

image distance cm
image orientation
magnification

(c)
p = 31 cm

image distance cm
image orientation
magnification

1/p + 1/q = 1/f

Insert p for each and solve for q
magnification is -q/p and orientation is upright or invert based on mag

To determine the image distance, image orientation, and magnification for each object distance, we can use the mirror equation and magnification formula.

The mirror equation is given by:

1/f = 1/p + 1/q

where f is the focal length of the mirror, p is the object distance, and q is the image distance.

For a concave mirror, the focal length (f) is half the radius of curvature (R):

f = R/2

Given that the radius of curvature (R) is 18 cm, the focal length is:

f = 18 cm / 2 = 9 cm

Now let's solve each part of the question:

(a) For p = 9 cm:

Using the mirror equation:

1/9 = 1/9 + 1/q

Simplifying the equation:

1/q = 1/9 - 1/9

1/q = 0

Since 1/q = 0, the image distance (q) approaches infinity.

Therefore, the image is formed at infinity.

Since the image is formed at infinity, it is a virtual image.

Since the image is virtual, it will be upright.

The magnification (m) can be determined using the formula:

m = -q / p

Substituting the values:

m = -∞ / 9

The magnification is -∞ (negative infinity).

(b) For p = 18 cm:

Using the mirror equation:

1/9 = 1/18 + 1/q

Simplifying the equation:

1/q = 1/9 - 1/18

1/q = 1/18

Multiplying both sides by 18:

18/q = 1

q = 18 cm

The image distance (q) is 18 cm.

Since the image distance is positive, the image is real.

Using the magnification formula:

m = -q / p

Substituting the values:

m = -18 / 18

m = -1

The magnification is -1.

(c) For p = 31 cm:

Using the mirror equation:

1/9 = 1/31 + 1/q

Simplifying the equation:

1/q = 1/31 - 1/9

1/q = 3/279

Multiplying both sides by 279:

279/q = 3

q = 93 cm

The image distance (q) is 93 cm.

Since the image distance is positive, the image is real.

Using the magnification formula:

m = -q / p

Substituting the values:

m = -93 / 31

m = -3

The magnification is -3.

Summary:
(a) Image distance: Infinity
Image orientation: Virtual
Magnification: Negative infinity

(b) Image distance: 18 cm
Image orientation: Real
Magnification: -1

(c) Image distance: 93 cm
Image orientation: Real
Magnification: -3

To find the image distance, orientation, and magnification for each object distance, we can use the mirror equation:

1/p + 1/q = 1/f

where p is the object distance, q is the image distance, and f is the focal length.

Given that the radius of curvature (R) is 18 cm, the focal length (f) can be calculated using the formula:

f = R/2

Substituting the value of R = 18 cm, we get:

f = 18/2 = 9 cm

Using the mirror equation, we can find the image distance (q) for each object distance (p) given:

(a) p = 9 cm:
Substituting p = 9 cm and f = 9 cm in the mirror equation:

1/9 + 1/q = 1/9

Simplifying, we get:

1/q = 1/9 - 1/9 = 0

Since 1/q = 0, q = infinity.
The image distance is infinity, indicating that the image is formed at infinity.

The image orientation for a concave mirror is inverted (upside down).

To find the magnification (m), we can use the formula:

m = -q/p

Substituting q = infinity and p = 9 cm, we get:

m = -infinity
The magnification is negative infinity, indicating an infinitely magnified inverted image.

(b) p = 18 cm:
Using the same steps as above, substituting p = 18 cm and f = 9 cm in the mirror equation, we get:

1/18 + 1/q = 1/9

Simplifying, we have:

1/q = 1/9 - 1/18 = 1/18

q = 18 cm

The image distance is 18 cm.

The image orientation for a concave mirror is inverted (upside down).

To find the magnification (m), we can use the formula:

m = -q/p

Substituting q = 18 cm and p = 18 cm, we get:

m = -18/18 = -1

The magnification is -1, indicating a real inverted image with the same size as the object.

(c) p = 31 cm:
Using the same steps as above, substituting p = 31 cm and f = 9 cm in the mirror equation, we get:

1/31 + 1/q = 1/9

Simplifying, we have:

1/q = 1/9 - 1/31

Finding the least common denominator (LCD) of 9 and 31, we have:

1/q = (31 - 9)/(279)

Simplifying, we get:

1/q = 22/279

q = 279/22 ≈ 12.68 cm

The image distance is approximately 12.68 cm.

The image orientation for a concave mirror is inverted (upside down).

To find the magnification (m), we can use the formula:

m = -q/p

Substituting q = 12.68 cm and p = 31 cm, we get:

m = -12.68/31 ≈ -0.41

The magnification is approximately -0.41, indicating a reduced real inverted image.