My friend has a Tamron 24mm lens on his Nikon D5000 and he says that because of its age and the fact it is manual focus, it is still actually a 24mm on his crop sensored camera. Is this right? If so, which 50mm lenses would still be 50mm on the Nikon d5100?
Your friend is right that it is actually always a 24mm lens — that is a property of the optics and never changes. But, he's wrong in saying that the crop factor does not apply. That's a property of the sensor size of the camera.
From a practical point of view, zoom — changing focal length — and cropping are interchangeable. So, using a camera with a smaller sensor (a "cropped" sensor) is effectively like using a longer focal length in several important ways, the most important of which is the field of view.
If, using the same camera, you take a photo with a 24mm lens and a 36mm lens, but crop the center 2/3rds from the 24mm photo and blow it up to the same size as a print of the 36mm image, the two photos will look virtually identical. (The blown-up image will have little more blur, of course — that's why we have different focal length lenses instead of just cropping. And in the real world there will be other unavoidable differences.)
When you mount the 24mm lens on a film camera, it has a horizontal field of view of about 74°. When you mount it on the D5000, it still projects the same image, but you only get the center portion, because the sensor is only that big, leaving you with a field of view of about 53°. That's the crop factor.
See What is "angle of view" in photography? for more on this — my answer there has an illustration which explains how this works.
In any case, the net effect of this is thankfully simple: the focal length of any lens, ancient or brand new, is an inherent property of the lens. It doesn't matter when it was made or what camera it was made for. If a camera has a sensor smaller than 35mm film, the crop factor can be applied to get an idea of the focal length which would give the same view on that format. That factor is also non-magical: because the sensor size of a given camera doesn't change, the crop factor always applies no matter what lenses you attach.
Another answer mentioned the coverage of the image circle. This is the circle of light projected from the back of the lens, and it is true that lenses designed for smaller sensors sometimes don't cover a full-frame sensor. That lets them be smaller, lighter, and cheaper, but has nothing to do with crop factor.
Note: the only times I've ever seen number printed on the lens or in specs "pre-converted" to 35mm-equivalent (by applying the crop factor to the real number) is on cameras with non-removable lenses. It's incredibly common in cameraphone specifications, and you'll often see it for superzoom compacts. That's probably mostly because the bigger numbers sound more exciting, not in an effort to be more useful. But I've never seen an interchangeable-lens system use anything but the real, physical numbers.
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You have already chosen your answer but I'd thought I would elaborate a little.
Yes, it stays the same! The age of the lens doesn't matter regarding focal length. As long as there is some way of fitting any lens onto a body (generally a bigger format lens on a smaller format body), whether by an adaptor or by physically changing the mount, the focal length always stays the same. Looking at this picture:
Looking at the red and blue lines, the focal length is the same. However, the format size changes. This is where the crop factor comes in.
Yes, the crop factor still applies! BUT! I want to stress that the term equivalent isn't necessarily the correct the correct term to use, it's just the most widely used. It should be RFoV or RAoV (Relative Field/Angle of View. This is my personal opinion, BTW.)
So, say the focal length is 24mm in that first diagram, that lens has a potential of seeing the red view, but will only see the blue view on a DX body so the image is cropped.
DX has a crop factor of 1.5 (generally. It varies). Any lens on it, times it by 1.5 and that will give the the relative field of view on a FF (35mm, 135 format) body. That's to say that this 24mm will still give you a similar shot than a 36mm on a FF, at the same distance. Here is another diagram I made up:
Please note that measurements are not to scale, only the ratio.
To further reduce confusion, my diagrams are only regarding angle of view (as stated in them and also as mattdm has linked to).
The crop factor is only relative to the 135 film format, which has dimensions 36mm x 24mm. This has a CF of 1. To make an image appear in the same angle of view as this format to a different sized format, you multiply the focal length by it's CF.
You have two cameras side by side looking at the same subject at the same working distance, one is FF and the other is APS. The FF will have a 36mm lens on it and the APS will have a 24mm lens on it, they will essentially see the same thing. That's what I'm demonstrating in my second diagram.
If you have both those cameras, both with the same focal length lens on it, the APS camera will need to be physically moved closer to the subject to have the same angle of view.
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Your friend is not right. A crop factor camera has a smaller sensor (in a way, you can think to a larger image to which a crop is applied). This cannot physically change when you put a lens, old or new, in front of it. So the lens projects an image on the sensor plane and you cut a crop of this image, thus you have a small field of view equivalent to those of a lens with a longer focal. In the same way for medium format camera the field of view is larger, since the sensor (the negative, for film) is larger than 24*35 mm squared.
You may want to consider larger-format lenses as well.
I'm pulling numbers out of thin air, here, more to illustrate the point than to be technically 100% correct.
On a medium format camera, 200 mm might be considered a "normal lens" in terms of focal length. This is because when it projects the image onto the film or sensor, the resulting image has a field of view (FoV) angle that is similar to what you'd get by projecting from a 50 mm lens onto a "full frame" (24x36 mm, traditionally called small-format) sensor.
Ignoring mechanics and electronics, focusing (no pun intended) instead squarely on optics, if you were to put that 200 mm lens on an APS-C body, you'd get a much more narrow field of view because only a small portion of the center of the image circle is actually used. You'd get something like the equivalent field of view as if you put a 1.6 * 200 mm * (200 mm / 50 mm) = 1300 mm lens on the MF body. (A 1300 mm lens on an APS-C body would be a massive telephoto lens.)
It's still the same lens, though.
If you put a 50 mm "full-frame" lens on that same MF body, if it somehow filled the MF body's entire image circle (which it won't, any more than an EF-S lens does on a full-frame body), it would give a similar field of view as a 12 mm lens does on a full-frame body (because in our hypothetical example, the MF body needs 4x the focal length to get the same field of view).
"Crop factor" is mostly marketing, and certainly not optics. The focal length of a lens doesn't change based on the size of the image recording material (sensor or film) in the camera you mount it on. The apparent field of view can change because some information is discarded, however.
Crop factor applies to every lens not specifically designed for crop sensors/film, and, I think, even most of those that are.
Focal length is a property of the optics of the lens, and so all lenses have the same properties when it comes to focal length on crop sensors, a 30 year old 10mm lens will appear to be a 15mm lens on your friends D5000, as will a 1 year old 10mm lens. Age plays no role in how the apparent focal length changes. With, for example, Canon EF-S lenses, they do not project an image circle big enough for a full frame camera, but their focal length is still their true focal length, NOT the focal length that it would appear to be when the image is taken.
The focal length of the lens appears to change because the size of the sensor means that only the center of the image circle is recorded. This can be compared to, in post processing, cropping out the center of an image to create an image that has a narrower field of view, but instead of being done in post, it is simply done via the sensor size.
Macro photography provides a good way to demonstrate this. If I take a 36mm*24mm piece of paper and drew a 23.6mm*15.6mm rectangle in the middle, then take a photo with a full frame camera and macro lens at 1:1 and also an APS-C* camera with same lens at 1:1, the full frame image can be cropped to the same size as the APS-C image and they will look exactly the same, the APS-C camera just does the cropping without any need for post.
Regarding the manual focus, this doesn't effect anything at all, because it only changes the fact that you need to manually turn the focus ring and that auto focus only adds a motor to do this automatically.