Unit 1: Reflection and Mirrors
Problem Set B
Overview:
Problem Set B targets your ability to use the mirror equation in order to solve relatively difficult word problems involving the location and size of images formed by concave and convex mirrors. There are 15 problems which involve multiple steps and a solid understanding of the sign conventions pertaining to the variables of the mirror equation. Many problems will involve the simultaneous use of two equations with two unknown values. Some problems will involve the expression of numerical information with messy or inconsistent units. You will use the two equations shown below:
1 / f = 1/ d_{o} + 1 / d_{i} 
M = h_{i} / h_{o} =  d_{i} / d_{o} 
In the above equations, the variable d_{o} represents the object distance or the distance between the mirror surface and the object. The variable d_{i} represents the image distance or the distance between the mirror surface and the image. The variable h_{o} represents the object height and the variable h_{i} represents the image height. The variable f stands for the focal length of the mirror. The variable M stands for the magnification of the image; it represents how many times bigger the image is than the object. In some problems, the focal length is not stated; rather, the radius of curvature of the spherical mirror is stated. The radius of curvature (R) is simply twice the focal length value (R = 2•f).
In Set B, the above equations are used to assist in the solution of physics word problems. Values of two or more quantities are typically expressed in the problem; and the goal of the word problem is to solve for an unknown quantity. Substitutions of known quantities will have to be made into the above equations and proper algebraic manipulations must be performed in order to solve for the unknown quantity. In some instances, the mathematics is difficult. In other instances, the problem is difficult because of issues regarding the +/ signs on quantities found in the two equations above or because of details regarding the units used to express given information.
Developing Effective ProblemSolving Habits
Using effective problemsolving habits will be critical as you approach these problems. Establishing good habits early in the course will pay tremendous dividends as more difficult problems appear in later sets and later units. Such habits include the following:
Perhaps one of the most problematic areas of this problem set is the task of dealing with the sign conventions associated with the image distance, focal length and image height. The table below summarizes the sign conventions associated with these quantities.

Sign Convention 

For our purposes, the object distance (d_{o}) will always be positive. 

For our purposes, the object height (h_{o}) will always be positive. 

A positive image distance (d_{i}) corresponds to an image location on the same side of the mirror as the object. A negative image distance (d_{i}) corresponds to an image located behind the mirror. 

A positive image height (h_{i}) corresponds to an upright image. A negative image height (h_{i}) corresponds to an inverted image. Since all upright images (positive h_{i} values) are virtual images located behind the mirror; upright images will thus be virtual images with negative image distances. Likewise, inverted images with their negative h_{i} values are real images that have positive image distance values. 

A concave mirror will have a positive focal length (f) and a convex mirror will have a negative focal length (f). 

Magnification values are positive whenever image heights (h_{i}) are positive. Thus, positive M values correspond to upright, virtual images located behind the mirror surface. And negative M values correspond to inverted, real images located on the object's side of the mirror. 
When reading a problem, give attention to cues within the problem in order to determine the sign on a given quantity. For instance, problem #3 describes an image being located a stated distance from a curved mirror. The stated value is simply the absolute value of the image distance. Whether the d_{i} value is positive or negative depends upon whether the image is in front of or behind the mirror. A careful reading of the problem statement along with an understanding of the sign convention for image distance (as stated in the table above) allows one to make the decision about the sign on d_{i}. These types of decisions are critical to your success on these problems. Making the correct decisions has nothing to do with your mathematical skills; rather, they are tests of your conceptual understanding and your willingness to read a problem carefully and give attention to details which may be important.
Solutions for Two Simultaneous Equations:
There will be many instances in these 15 problems in which you will be asked to solve for an unknown quantity (such as d_{i}) using the mirror equation but both the other two quantities are not stated. Such problems usually have a statement of the effect: "the image is real and three times the size of the object." Such a statement reveals information about the magnification of the image. Size and height can be treated synonymously. Stating that the image is three times the size of the object is stating that the ratio h_{i}/h_{o} is ±3. Determining whether h_{i}/h_{o} is +3 or 3 demands an understanding of the sign conventions. The h_{o} value is always positive (for our purposes). The h_{i} value is positive for upright images and negative for inverted images. Since this statement asserts that the image is real (and thus inverted), a 3 value must be assigned to the h_{i}/h_{o} ratio. Since h_{i}/h_{o} is equal to d_{i}/d_{o}, the 3 value can be equated with d_{i}/d_{o}. From this, an equation can be written relating the value of d_{i} to the value of d_{o}. This expression for d_{i} in terms of d_{o} can be substituted into the mirror equation in order to transform it into a single equation with a single unknown. Customary algebraic manipulations can then be performed in order to solve for d_{i}.
The table below summarizes the process of transforming a verbal statement into a mathematical equation which ultimately is used to substitute into the mirror equation.






Additional Readings/Study Aids:
The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the proper use of the mirror equation and magnification ratio and in your ability to make the connection between the image characteristics and the proper sign (+ vs. ) for stated values.
View Sample Problem Set.

Description 


Calculation of d_{i} requiring an understanding of the fR relation. 


Referring to the previous problem; routine calculation of M. 


Calculation of d_{o}; requires conceptual decisionmaking with regards to sign conventions. 


Moderately difficult twostep problem requiring significant thinking, and some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


A twostep problem involving the calculation of M; requires conceptual decisionmaking with regards to sign conventions. 


A multistep problem involving the calculation of do; requires some conceptual decisionmaking with regards to sign conventions and the use of two equations and two unknowns. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


A twostep problem involving the calculation of f; requires conceptual decisionmaking with regards to sign conventions. 


A twostep problem involving the calculation of f; requires conceptual decisionmaking with regards to sign conventions. 


A twostep problem involving the calculation of f; requires conceptual decisionmaking with regards to sign conventions. 

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