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Chapter 30: Problem 7

The energy at which the fundamental forces are expected to unify is about \(10^{19}\) GeV. Find the mass (in kilograms) of a particle with rest energy \(10^{19} \mathrm{GeV}\).

Short Answer

Expert verified
Answer: The mass of a particle with rest energy \(10^{19} \mathrm{GeV}\) is approximately \(1.783 \times 10^{-28} \: kg\).

Step by step solution

01

Convert the energy from GeV to Joules

To convert the energy given in GeV to Joules, we will use the conversion factor mentioned above. We have the energy \(E = 10^{19} \mathrm{GeV}\), and the conversion factor \(1\: GeV = 1.602176634 \times 10^{-10}\: J\). Multiplying the energy in GeV by the conversion factor gives us the energy in Joules: \(E = 10^{19} \mathrm{GeV} \times 1.602176634 \times 10^{-10} \: J/\mathrm{GeV}\)
02

Calculate the mass using Einstein's equation

Now that we have the energy in Joules, we can use Einstein's equation \(E=mc^2\) to find the mass. Rearrange the equation to solve for the mass: \(m = \frac{E}{c^2}\) Where \(E\) is the energy in Joules (calculated in Step 1) and \(c\) is the speed of light, approximately equal to \(2.998\times10^8 \: \mathrm{m/s}\). Substitute the values and calculate the mass: \(m = \frac{10^{19} \mathrm{GeV} \times 1.602176634 \times 10^{-10} \: J/\mathrm{GeV}}{(2.998\times10^8 \: \mathrm{m/s})^2}\)
03

Evaluate and find the mass in kilograms

Finally, we will evaluate the expression to find the mass in kilograms: \(m = \frac{10^{19} \times 1.602176634 \times 10^{-10}}{(2.998\times10^8)^2} \: kg\) \(m \approx 1.783 \times 10^{-28} \: kg\) So, the mass of a particle with rest energy \(10^{19} \mathrm{GeV}\) is approximately \(1.783 \times 10^{-28} \: kg\).

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Most popular questions from this chapter

A sigma baryon at rest decays into a lambda baryon and a photon: $\Sigma^{0} \rightarrow \Lambda^{0}+\gamma .$ The rest energies of the baryons are given by \(\Sigma^{0}=1192 \mathrm{MeV}\) and \(\Lambda^{0}=1116 \mathrm{MeV} .\) What is the photon wavelength? [Hint: Use relativistic formulas and be sure momentum is conserved as well as energy.]
Show that the charge of the neutron and the charge of the proton can be derived from their constituent quark content.
When a proton and an antiproton annihilate, the annihilation products are usually pions. (a) Suppose three pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (b) Suppose five pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (c) What is the maximum number of pions that could be produced if the kinetic energies of the proton and antiproton are negligibly small? The mass of a charged pion is \(0.140 \mathrm{GeV} / c^{2}\) and the mass of a neutral pion is \(0.135 \mathrm{GeV} / c^{2}.\)
Which fundamental force is responsible for each of the decays shown here? [Hint: In each case, one of the decay products reveals the interaction force.] (a) \(\pi^{+} \rightarrow\) \(\mu^{+}+v_{\mu},\) (b) $\pi^{0} \rightarrow \gamma+\gamma(\mathrm{c}) \mathrm{n} \rightarrow \mathrm{p}^{+}+\mathrm{e}^{-}+\bar{v}_{\mathrm{e}}$
A pion (mass \(0.140 \mathrm{GeV} / c^{2}\) ) at rest decays by the weak interaction into a muon of mass \(0.106 \mathrm{GeV} / c^{2}\) and a muon antineutrino: \(\pi^{-} \rightarrow \mu^{-}+\nabla_{\mu}\) Ignoring the rest energy of the antineutrino, what is the total kinetic energy of the muon and the antineutrino?
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