Reflection and Mirrors Legacy Problem #12 Guided Solution

In physics, a good problem solver reads the problem carefully and begins to identify the known values, equating them with variables or symbols that are used in physics equations. Then begins to identify the unknown quantities, again equating them with symbols that are commonly found in physics equations, and then plots out a strategy as to how to get from the known values to the unknown values. Here in this problem, we're given three bits of given information. We're given the height of a light bulb, and I usually call that HO. That's the object height. And we're given the distance that the light bulb is from a spherical concave mirror, and I call that DO for object distance. So HO is equal to 56 millimeters, and DO is equal to 124 centimeters. And then the focal length is given, and that's what I call F. It's equal to 62 centimeters. My two unknowns in part A are the image distance, DI, and the image height, which I call HI for HI. Now, what I wish to do is find an equation that relates these variables to one another, and the two options are the mirror equation and the magnification equation. The mirror equation goes 1 divided by DO plus 1 divided by DI equal 1 divided by F. And since I know DO and F, I can use this equation to solve for the image distance for DI. So I rearrange the equation to form 1 over DI equal 1 over F minus 1 over DO. And then I substitute values of 62 for F and 124 for DO, and I evaluate the right side of the equation. It evaluates to 8.0645 times 10 to the negative third. That's not the answer, rather, that's 1 divided by the answer, or 1 divided by DI. If I take the reciprocal of the left side, I get DI. If I take the reciprocal of the right side, I get 124 centimeters. So the image distance, DI, is 124 centimeters, and that leaves the image height remaining to be solved for. So to solve for the image height, I use the equation, HI per hole equal negative DI per DO. That's HI divided by HO is equal to the negative of DI divided by DO. Now I know everything in that equation except for I. So I can substitute in and solve for I. Substituting gives me the following. HI divided by 56 millimeters is equal to the negative of the ratio, 124 centimeters divided by 124 centimeters. If I solve this for HI, I end up getting negative 56 millimeters. So now I've solved for part A questions. DI equal 124 centimeters, and HI equal negative 56 centimeters. Now in part B, I'm asked to describe the orientation and type of the image. The orientation of this image is that it is upside down, or inverted. I know this because the HI image height turned out to be negative, and for the type of the image, it's going to be a real image. I know this because the image distance ended up being a positive value, placing it on the same side of the mirror as the object itself. In other words, when you place the object at a distance twice the focal length, you end up getting an image which is exactly the same size, only inverted and upside down, and it's a real image.