Have you ever wonder how come lens equivalency works across different digital formats? Today I did a small illustration using the coffee mixing sticks.
In this illustration, I have tried to show this idea that if you want to achieve the same angle of view across full frame and crop sensor then how and why you need different focal length. Why 50mm on full frame gives an equivalent angle of view of 75mm on the crop sensor. The answer is, to cover the large area of the sensor (which will be FF in this case) you need a bigger circle of projection which means you ultimately have to design lenses of a smaller focal length (50mm in this case) which ultimately covers the full frame.
In case of a crop sensor, you have less area to cover which means the small angle of projection you need to cover the entire sensor which ultimately achieves by designing the longer focal length lenses. Longer focal length produces a small angle of projection.
Hope this simple illustration will clear this confusion that why 50mm on Full Frame is equivalent to 75mm on a crop sensor. Also, note that through this illustration, physically focal length has nothing to go with the sensor size. You can switch the focal length and results from the 75mm will be 75mm and 50mm will act as a 50mm. If you switch 50mm on a crop sensor, it will remain 50mm and will give you the same depth of field of 50mm because physically the focal length does not change so this non-sense idea which circulating around the photography world that if you use Full Frame lenses on a Crop sensor camera bodies "multiplies" your focal length and gives you extra reach .... is total NON-SENSE.