Force to keep photon in circular motion

[quietrain]

Homework Statement

A proton has an energy of 2.5 × 10−10 J. What force is needed to keep it going in
a circle having a 1.0 km radius

The Attempt at a Solution

so i tried to find v first.

total energy E = 2.5 x 10^-10 J =$$\gamma$$ mc²

so m = photon mass = 1.67 x 10^-27

so i found $$\gamma$$ = 1.66
and v = 0.798c

so when i sub into f = mv²/R , i get f = 9.57 x 10^-14

but the answer is 1.6 x 10^-13

which means that the photon speed is roughly 0.9++ c

so anyone knows what's wrong?

btw why can't i use E= hf, get f, convert to T =1/f, then use circumference / T = v ?
i realize that v i get is faster than the speed of light. but which step is wrong in this attempt?

thanks!

 

[quietrain]

hmm.. i realize that i have to use relativistic mass in this case. to get the answer.

so basically anything that has speed comparable to c, we have to use relativistic mass?

so gamma was 1.66, so it means that as the photon moves with speed close to light, its mass increases? 1.66m ? so is this correct? speed increase = mass increase? from E = mc ^2btw, anyone knows what's wrong with my 2nd type of attempt in the first post?