Our camera won't turn on despite a fully charged new battery?

Our camera won't turn on despite a fully charged new battery?
Added (1). Our camera won't turn on despite a fully charged new battery?

Edit

Our Nikon S5200 camera won't turn on despite a fully charged new battery

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What alternate lenses will work for my Nikon D3100?

I am searching on eBay for a cheaper lens to play with for my D3100. I don't know what to look for.

I believe first, I look for the lens having an "F Mount", do I have to look for a thread size or does that just matter when looking for filters? What should I be looking for? What is definitely not compatible?

I do shoot a lot in manual so I don't mind about AF. I am looking for a cheap lens.
Added (1). I am interested in prime lenses. I am not too keen on looking for adapters.

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35mm vs 50mm on crop sensor DSLR?

I'm using a Nikon D3300 and looking at the Nikon 35mm 1.8G or the Nikon 50m 1.8G. Not sure what to get?
I currently have a sigma 17-50mm 2.8 lens and a Nikon 70-300mm telephoto lens

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Internally zooming/focusing full frame Nikon lenses?

I'm looking for a general good walkaround full frame Nikon F mount lens which is weather sealed and focuses/zooms internally. I was originally looking at the 24-70 2.8, but it externally zooms. I am aware of the Sigma 24-35mm f2 which zooms and focuses internally, but isn't weather sealed.

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