Modern Physics
Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 1, Problem 14P

A rod of length L0 moves with a speed v along the horizontal direction. The rod makes an angle of θ0 with respect to the x′-axis. (a) Show that the length of the rod as measured by a stationary observer is given by L = L 0 [ 1 ( v 2 / c 2 ) cos 2 θ 0 ] 1 / 2 . (b) Show that the angle that the rod makes with the x-axis is given by the expression tan θ = γ tan θ0. These results show that the rod is both contracted and rotated. (Take the lower end of the rod to be at the origin of the primed coordinate system.)

(a)

Expert Solution
Check Mark
To determine

To show that the length of the rod as measured by a stationary observer is given by L=L0[1(v/c)2cos2θ0]1/2

Answer to Problem 14P

The length of the rod as measured by a stationary observer is given by L=L0[1(v/c)2cos2θ0]1/2 is shown.

Explanation of Solution

The rod of length L0 moves along the horizontal axis and hence the horizontal component of the rod gets contracted.

The horizontal component of the rod is given by,

Lx=L0cosθ0

Here, Lx is the horizontal component of the rod and θ0 is the angle made by the rod.

The horizontal component of the rod as measured by the stationary observer is,

Lx=Lx[1(v/c)2]1/2=L0cosθ0[1(v/c)2]1/2

Here, Lx is horizontal component of the rod as measured by the stationary observer

The vertical component of the rod is given by,

Ly=L0sinθ0

Here, Ly is the vertical component of the rod.

The vertical component of the rod as measured by the stationary observer is,

Ly=Ly[1(v/c)2]1/2=L0sinθ0[1(v/c)2]1/2=L0sinθ0[1(0/c)2]1/2(v=0 along vertical direction)=L0sinθ0

Here, Ly is the vertical component of the rod as measured by the stationary observer.

Conclusion:

The length of the rod as measured by a stationary observer is L=[(Lx)2+(Ly)2]1/2=[(L0cosθ0[1(v/c)2]1/2)2+(L0sinθ0)2]1/2=L0[cos2θ0(1v2c2)+sin2θ0]1/2=L0[1v2c2cos2θ0]1/2

Therefore, the length of the rod as measured by a stationary observer is given by L=L0[1(v/c)2cos2θ0]1/2 is shown.

(b)

Expert Solution
Check Mark
To determine

To show that the angle the rod makes with the x axis is tanθ=γtanθ0

Answer to Problem 14P

The angle the rod makes with the x axis is tanθ=γtanθ0 is shown.

Explanation of Solution

The angle the rod makes with the x axis is,

tanθ=LyLx=L0sinθ0L0cosθ0/γ(γ=[1(v/c)2]1/2)=γtanθ0

Conclusion:

Therefore, the angle the rod makes with the x axis is tanθ=γtanθ0 is shown.

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Chapter 1 Solutions

Modern Physics

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