Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Question
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Chapter 30, Problem 28P

(a)

To determine

Find the speed of proton.

(a)

Expert Solution
Check Mark

Answer to Problem 28P

The speed of proton is 0.999999991c.

Explanation of Solution

The proton is accelerated energy is 7 TeV.

Write the equation for relativistic energy.

E=mc21v2c2

Here, E is the energy proton, m is the mass of proton, v is speed of proton, and c is the speed of light.

Rearrange the above equation to get the v.

v=c1(mc2)2E2 (I)

Conclusion:

Substitute 7 TeV for E and 398.272 MeV for mc2 in equation I.

v=c1(398.272 MeV)2(7 TeV)2=0.999999991c

Therefore, the speed of proton is 0.999999991c.

(b)

To determine

Find time it would take proton to make one rotation.

(b)

Expert Solution
Check Mark

Answer to Problem 28P

The time taken by proton to make one rotation is 12 ns.

Explanation of Solution

The circumference of tunnel is 27 km.

Write the relativistic equation to find time take by proton.

t=L0c1v2c2 (II)

Here, the t is the time taken by proton and L0 is the circumference of the tunnel.

Conclusion:

Substitute 27 km for L0, 3×108 m/s for c  and 0.999999991c for v in equation II.

t=27 km3×108 m/s1(0.999999991c)2c2=27×103 m3×108 m/s(1.3416×104)=0.0012×105 s=12 ns

Therefore, the time taken by proton to take on rotation is 12 ns.

(c)

To determine

Find time it would take proton to make one rotation in the reference frame.

(c)

Expert Solution
Check Mark

Answer to Problem 28P

The time taken by proton to make one rotation in reference frame is 90 μs.

Explanation of Solution

Write the equation for time taken by proton in reference frame.

t0=L0c (III)

Here, t0 is the time taken by proton to take one rotation in reference frame.

Conclusion:

Substitute 27 km for L0 and 3×108 m/s for c in equation III.

t0=27 km3×108 m/s=90μs

Therefore, the time taken by proton to make one rotation in reference frame is 90 μs.

(d)

To determine

Find the de Broglie wavelength of proton.

(d)

Expert Solution
Check Mark

Answer to Problem 28P

The de Broglie wavelength of proton is 1.8×1019 m.

Explanation of Solution

Write the relativistic de Broglie equation.

λ=hmv1v2c2 (IV)

Here, λ is de Broglie wavelength and h  is the Plank’s constant.

Conclusion:

Substitute 0.999999991c for v, 6.626×1034 Js for h, 1.673×1027 kg for m and 3×108 m/s for c in equation IV.

λ=6.626×1034 Js(1.673×1027 kg)(0.999999991)(3×108 m/s)1(0.999999991c)2c2=1.8×1019 m

Therefore, the de Broglie wavelength of proton is 1.8×1019 m.

(e)

To determine

Find the mosquito’s speed.

(e)

Expert Solution
Check Mark

Answer to Problem 28P

The mosquito’s speed is 1.5 m/s.

Explanation of Solution

First the decision has to be made whether to use relativistic or nonrelativistic formula.

Write the kinetic energy in joules,

7 TeV=(7×1012 eV)(1.602×1019 J/eV)=1.1 μJ

The rest energy of mosquito is given below,

E0=mc2=(1.0×106 kg)(3.0×108 m/s)2=9.0×1010 J

Therefore, the rest energy of mosquito is much higher than the kinetic energy, thus the nonrelativistic equation must be used to calculate the speed.

Write the nonrelativistic equation for speed.

K=12mv2

Here, K  is the kinetic energy, m  is the mass of mosquito and v is speed of mosquito.

Rewrite the above equation to find speed.

v=2Km (V)

Here, λ is de Broglie wavelength and h  is the Plank’s constant.

Conclusion:

Substitute 1.1 μJ for K and 1.0×106 kg for m in equation V.

v=2(1.1 μJ)1.0×106 kg=2.2×106 kgm2/s21.0×106 kg=1.5 m/s

Therefore, the mosquito’s speed is 1.5 m/s.

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