Question:

A concave mirror with a radius of curvature of 20 cm is used to form an image of an object. The object is placed 35 cm in front of the mirror.

(a) Determine the position and magnification of the image formed by the mirror.

(b) Is the image virtual or real? Is it upright or inverted?

(c) Calculate the focal length of the mirror.

Answer:

(a) To determine the position and magnification of the image formed by the mirror, we can use the mirror equation:

1/f = 1/do + 1/di

where f is the focal length of the mirror, do is the object distance (the distance from the object to the mirror), and di is the image distance (the distance from the image to the mirror).

Given: Radius of curvature (R) = -20 cm (concave mirror) do = -35 cm (negative since the object is placed in front of the mirror)

First, let's calculate the image distance (di):

1/f = 1/do + 1/di

=> -1/20 cm = 1/-35 cm + 1/di

Multiplying through by (-35 * di):

-di + (-35 * -35) = -20 * (-35)

-di + 1225 = 700

-di = 700 - 1225

-di = -525

di = 525 cm

The image distance (di) is -525 cm, which means it is located on the same side as the object (in front of the mirror).

To find the position of the image, we can use the formula:

magnification (m) = -di/do

Substituting the given values:

m = -525 cm / -35 cm = 15

The magnification of the image is 15.

(b) To determine whether the image is virtual or real, and whether it is upright or inverted, we can analyze the sign of the magnification.

Since the magnification (m) is positive (+15), the image is upright. And since do and di have opposite signs, the image is virtual.

(c) The focal length of a concave mirror can be calculated using the formula:

1/f = 1/do + 1/di

Substituting the given values:

1/f = 1/-35 cm + 1/525 cm

Multiplying through by (-35 * 525):

525 - 35f = -35 * 525

525 - 35f = -18375

-35f = -18900

f = -18900 / -35

f ≈ 540 cm

The focal length of the concave mirror is approximately 540 cm.

Therefore, the answers to the questions are:

(a) The image is formed at -525 cm from the mirror, with a magnification of 15. (b) The image is virtual and upright. (c) The focal length of the mirror is approximately 540 cm.