Question

Physicists and engineers from around the world have come together to build the largest accelerator in the world, the Large Hadron Collider (LHC) at the CERN Laboratory in Geneva, Switzerland. The machine accelerates protons to high kinetic energies in an underground ring 27 km in circumference.
a. What speed v of proton in the LHC if the proton's kinetic energy is 6.7 TeV? (Because v is very close to c, write v=(1−Δ)c and give your answer in terms of Δ).
b. Find the relativistic mass, mrel, of the accelerated protons in terms of their rest mass.

Answer 1

Solution :

Energy of photon, E = 6.7 eV

                              E = $6.7 \times 1.602 \times 10^{-7}$ joule

Kinetic energy, $K.E. =\frac{1}{2} mv^2 = 1.602 \times 6.7 \times 10^{-7}$

$v^2=\frac{2 \times 1.602 \times 6.7 \times 10^{-7}}{1.6726 \times 10^{-27}}$

   $=12.834 \times 10^{-20}$

Kinetic energy at high speeds

$(r-1)\times mc^2 = 6.7 \ eV$

$(r-1)=\frac{6.7 \times 1.602 \times 10^{-7}}{1.6726 \times 10^{-27} \times 9 \times 10^{16}}$

r - 1 = 7130

r = 7130 + 1

r  = 7131

$\frac{1}{\sqrt{1-\frac{v^2}{C^2}}}=7131$

$1-\frac{v^2}{C^2} = \left(\frac{1}{7131}\right)^2$

$v^2=C^2\left[1-\left(\frac{1}{7131}\right)^2\right]$

$v=0.99999999017C$

Δ = 1 - 0.99999999017

   = 0.00000000933

Relative mass, $m_{rel}=r.m$

                                $=7131 \times 1.6728 \times 10^{-27}$

                               $=1.1927 \times 10^{-23}$ kg