Reflection and Mirrors Legacy Problem #16 Guided Solution
Problem*
A convex spherical mirror has a focal point located a distance of 24.6 cm from the surface of the mirror. (You will have to decide for yourself whether f is + or -.)
- Find the image distance (in cm) for an object distance of 76.8 cm.
- Determine the magnification of this image.
Audio Guided Solution
In this question, we're told that we have a convex mirror and that its focal point is located 24.6 centimeters from the mirror's surface. Now, a good problem solver is always conscious of concepts, never divorcing the problem, the mathematics of the problem, from the concepts. And so here, if I'm told that the focal point is 24.6 centimeters from the mirror, what I know is that the focal length is equal to negative 24.6 centimeters. After all, that focal point is on the side of the mirror, on the concave side of the mirror, the side opposite the side the object's on. Now, I'm told that the object's 76.8 centimeters from the mirror, and part A, I wish to calculate the image distance. So I use the mirror equation, which states that 1 over f, the focal length, is equal to 1 over do for object distance plus 1 over di for image distance. I can rearrange the equation to solve for image distance. It becomes 1 over di equal 1 over f minus 1 over do. And then I can substitute negative 24.6 centimeters in for f and 76.8 centimeters in for do. If I evaluate the right side of the equation, I end up getting negative 0.053671. And that's not my answer for di, and that's actually the reciprocal of di. Now, if I take the reciprocal of both sides, I end up with di equal negative 18.6320. The negative 18.6 is the answer when rounded to three significant digits, and the negative part of that answer simply indicates that the image is located behind the mirror. Now in part B, I wish to calculate the magnification. The magnification can be found one of two ways. First, you could calculate the high per hole ratio if you knew high, image height, and hole, object height. If you don't, you have to use the other means of calculating magnification, which is magnification equal, the negative di-do ratio. So here, the di value is what we found in part A. It's negative 18.6320. To divide that by the do value of 76.8 centimeters, and that gives me a value of 0.24260 rounded to three digits. It's 0.243. Now, that's a magnification that's a positive value, as we would expect for convex mirrors produce images which are upright, and thus have positive magnifications.
Solution
- di = -18.6 cm
- M = 0.243
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(\descriptive{d_o}{d_o,distance object} = 24.2\unit{cm}\); \(\descriptive{d_i}{d_i,distance image} = 16.8\unit{cm}\); \(\descriptive{f}{f,focal length} = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use. Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
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