Curved Mirror Calculations Study Guide

Learning Objectives: By the conclusion of the lesson, students will understand the methodology for utilizing curved mirror equations. This includes determining the height of a resulting image and its specific distance from the mirror surface.
Success Criteria: Students must demonstrate the ability to:
- Apply curved mirror equations to find the image height ( $h_i$ ).
- Apply curved mirror equations to find the image distance ( $d_i$ ) for both concave (converging) and convex (diverging) mirrors.

The Mirror Equation: This formula relates the focal length of the mirror to the distances of the object and the image from the mirror's vertex.
- $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
The Magnification Equation: This formula relates the ratio of the image height to the object height with the ratio of the image distance to the object distance.
- $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

$d_o$ (Object Distance): The distance measured from the object being reflected to the surface of the mirror. In standard calculations, this is always a positive value (d_o > 0).
$d_i$ (Image Distance): The distance measured from the mirror surface to the location where the image is formed.
$f$ (Focal Length): The distance from the mirror’s vertex to its focal point ( $F$ ).
$h_o$ (Object Height): The physical height of the object being placed in front of the mirror.
$h_i$ (Image Height): The calculated height of the image produced by the mirror.
$m$ (Magnification): A dimensionless value indicating how many times larger or smaller the image is compared to the object.

Term	Concave Mirror (Converging)	Convex Mirror (Diverging)
Focal length ( $f$ )	Positive (f > 0)	Negative (f < 0)
Object distance ( $d_o$ )	Always positive (d_o > 0)	Always positive (d_o > 0)
Image distance ( $d_i$ )	Positive for real images; negative for virtual images	Always negative (images are always virtual)
Magnification ( $m$ )	Positive for upright images; negative for inverted images	Always positive (images are always upright)
Image height ( $h_i$ )	Positive for upright images; negative for inverted images	Always positive (images are always upright)

Example 1: Finding image distance and size
- Scenario: A light bulb with a height of $4.00\,cm$ is placed $45.7\,cm$ from a concave mirror. The mirror has a focal length of $15.2\,cm$ .
- Given Data: $h_o = 4.00\,cm$ , $d_o = 45.7\,cm$ , $f = 15.2\,cm$ .
- Goal: Calculate the image distance ( $d_i$ ) and the image height ( $h_i$ ).
Example 2: Analyzing a specific object placement
- Scenario: An object with a height of $2.5\,cm$ is placed $40.0\,cm$ in front of a concave mirror that has a focal length of $12\,cm$ .
- Given Data: $f = 12\,cm$ , $h_o = 2.5\,cm$ , $d_o = 40.0\,cm$ .
- Goal: Calculate the image distance ( $d_i$ ) and the image height ( $h_i$ ).
Example 3: Object placed within the focal point
- Scenario: A $4.0\,cm$ tall light bulb is placed at a distance of $8.3\,cm$ from a concave mirror with a focal length of $15.2\,cm$ .
- Note: Because the object is placed between the focal point and the mirror surface (d_o < f), the resulting image will be virtual ( $d_i$ will be negative).
- Given Data: $h_o = 4.0\,cm$ , $d_o = 8.3\,cm$ , $f = 15.2\,cm$ .
- Goal: Determine the image distance ( $d_i$ ) and the image height ( $h_i$ ).

Primary Distinction: The fundamental difference in calculations for convex mirrors compared to concave mirrors is that convex mirrors always possess a negative focal length (f < 0).
Example 4: Convex mirror image determination
- Scenario: A convex mirror has a focal length of $-0.90\,m$ . An object with a height of $0.40\,m$ is positioned $2.5\,m$ from the mirror.
- Given Data: $f = -0.90\,m$ , $h_o = 0.40\,m$ , $d_o = 2.5\,m$ .
- Goal: Calculate the image distance ( $d_i$ ) and the image height ( $h_i$ ).

Extra Problem 1 (Concave Mirror):
- Scenario: A concave mirror has a focal length of $6.0\,cm$ . An object with a height of $0.60\,cm$ is placed $10\,cm$ in front of the mirror.
- Goal: Calculate the image distance ( $d_i$ ) and image height ( $h_i$ ).
Extra Problem 2 (Dancer's Mirror):
- Scenario: A dancer uses a concave mirror to apply makeup. The dancer's face is positioned $35\,cm$ in front of the mirror surface. The resulting image is found to be $72\,cm$ behind the mirror.
- Analysis: Because the image is behind the mirror, the image distance ( $d_i$ ) must be treated as a negative value ( $-72\,cm$ ).
- Goal: Use the mirror equation to calculate the focal length ( $f$ ) of the mirror.

Worksheet: Complete questions in Part 1 (numbers 1 through 5) on the mirror calculations worksheet.
Textbook Problems: Page 451, questions #1, #2, and #4.
Additional Work: Curved mirror equation worksheet.
Upcoming Assessments:
- Quiz on Images in Curved Mirrors: Scheduled for Friday (Lesson 10).
- Museum Security Project: Due Wednesday (Lesson 8).