You want to produce a mirror that can produce an upright image that would be twice as tall as the object when placed 5 cm in front of it. What shape should this mirror be? What radius of curvature should the mirror have?

Hey, everyone. Thank you for joining us here at Clutch Physics. Glad you can make it. Alright. So here we wanna design a mirror. We wanna produce an upright image that would be twice as tall as the object. So right off the bat, we know we're talking about magnification. So the magnification of this mirror, we want it to be 2. Right? We want it to be twice as tall as the object. Right? Because this is equal to as a side note, this is equal to the image height over the object height. Okay. That's equal to magnification. There's another thing that magnification is equal to, and that is the negative ratio of the image distance over the object distance. Okay? And this is gonna be important. Right? Because we wanna find the, focal length of this mirror. Right? Because it's asking us what shape the mirror should be. So we know that focal length is gonna tell us the shape. Right? Because if the focal length is greater than 0, then it's a concave mirror. If the focal length is less than 0, then we know it's convex. Okay. So positive focal length concave, negative focal length convex. Alright? And this is also gonna tell us the radius of curvature. Right? Because f is equal to r over 2. Okay? So that kinda lays out the roadmap. So we wanna find the image distance. Right? We already have object distance. It's sold to us. It's 5 centimeters. We wanna find the image distance and then use our mirror equation to find, the image distance, given the object distance. Okay? So, what is the image distance? Well, we can use we can find that using our magnification equation. Image distance is gonna be equal to negative two times object distance. We already said image, object distance is 5. So image distance is negative 10 centimeters. Okay? So we have a negative image distance. Okay? So we're gonna include that in our in our mirror equation, which is one over s object plus one over s image equals to one over f. Plugging this in, we get 1 over 5 minus Oops. See I made a mistake there. This is a minus 1 over 10. Right? Image distance is negative equal to one over f. This gives us a focal length of 10 centimeters, which means what? We have a concave mirror. Positive focal length, meaning we have a concave mirror. Alright? What is the radius of curvature? Again, this is gonna be f is equal to r over 2, which means that r is equal to 2 times f. So we have a radius of curvature of 20 centimeters. Alright? Those are our 2 answers. We have a 20 centimeter radius of curvature, and this is a concave mirror. Alright, guys. So that's it for this one. Thanks for watching.

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You want to produce a mirror that can produce an upright image that would be twice as tall as the object when placed 5 cm in front of it. What shape should this mirror be? What radius of curvature should the mirror have?

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