Reflection and Mirrors Legacy Problem #10 Guided Solution
Problem*
Obtaining a large spherical mirror with a focal length of 0.654 m from the Physics Storeroom, Mr. H takes his last period class outside for a fascinating demo. A student volunteer holds the mirror at an angle such that the face of the mirror is directed towards the Sun - roughly 1.46x1011 m away. Mr. H then uses a piece of paper with George Washington's picture on it to focus the image of the sun on the sheet of paper. Before the paper engulfs in flames, a bright image of the sun can be seen on the paper. Use the mirror equation to calculate the distance from the mirror to the image of the sun.
Audio Guided Solution
Here is a popular physics demonstration which involves taking a rather large spherical mirror outside and pointing it up towards the sun. A sheet of paper is then held near the mirror and slowly moved away from the mirror focusing the image of the sun on that sheet of paper. When the paper is placed at exactly the right location, the image location of the sun, the paper is slowly engulfed in flames. Now what we wish to do in this question is calculate the image distance, the distance from the mirror to this image location of the sun. We're given the object distance which is 1.46 times 10 to the 11th meters. We're given the focal length of the mirror. So we need to use the mirror equation which says 1 over object distance plus 1 over image distance equal 1 over focal length. And since we're trying to solve for the image distance, we wish to rearrange the equation to get the image distance termed by itself. It rearranges to 1 over di or 1 over image distance is equal to 1 over f or 1 over focal length minus 1 over do or 1 over object distance. So 1 over di equal 1 divided by 0.654 minus 1 divided by 1.46 e to the 11th. Now when we do our math for this, we evaluate the right side of the equation to be 1.5291. It's equal to 1 over the image distance. If you take the reciprocal on the left side, 1 over di, you get di itself. And if you take the image and if you take the reciprocal on the right side, 1 divided by 1.5291, you get 0.6540 or simply 0.654 meters as the answer. Now I can imagine a student doing this math and not getting nearly the numbers that I described. And if that's the case, my first bet is that when you enter the number 1.46 times 10 to the 11th, you're entering exactly as I mentioned. You have to be careful when you divide by that number because what you end up doing if you enter it wrong is you end up dividing by 1.46 and then multiplying by 10 to the 11th. You want to divide by the whole thing, 1.46 times 10 to the 11th. So make sure you either surround it by parentheses or you enter it into your calculator as 1.46 second comma on a graphing calculator, 11. And that would enter into your panel of your calculator, it would read 1.46 double e to the 11th. That should do it for you.
Solution
0.654 m
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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