Answer :
Answer:
From the question we are told that
The length of the rod is [tex]L_o[/tex]
The speed is v
The angle made by the rod is [tex]\theta[/tex]
Generally the x-component of the rod's length is
[tex]L_x = L_o cos (\theta )[/tex]
Generally the length of the rod along the x-axis as seen by the observer, is mathematically defined by the theory of relativity as
[tex]L_xo = L_x \sqrt{1 - \frac{v^2}{c^2} }[/tex]
=> [tex]L_xo = [L_o cos (\theta )] \sqrt{1 - \frac{v^2}{c^2} }[/tex]
Generally the y-component of the rods length is mathematically represented as
[tex]L_y = L_o sin (\theta)[/tex]
Generally the length of the rod along the y-axis as seen by the observer, is also equivalent to the actual length of the rod along the y-axis i.e [tex]L_y [/tex]
Generally the resultant length of the rod as seen by the observer is mathematically represented as
[tex]L_r = \sqrt{ L_{xo} ^2 + L_y^2}[/tex]
=> [tex]L_r = \sqrt{[ (L_o cos(\theta) [\sqrt{1 - \frac{v^2}{c^2} }\ \ ]^2+ L_o sin(\theta )^2)}[/tex]
=> [tex]L_r= \sqrt{ (L_o cos(\theta)^2 * [ \sqrt{1 - \frac{v^2}{c^2} } ]^2 + (L_o sin(\theta))^2}[/tex]
=> [tex]L_r = \sqrt{(L_o cos(\theta) ^2 [1 - \frac{v^2}{c^2} ] +(L_o sin(\theta))^2}[/tex]
=> [tex]L_r = \sqrt{L_o^2 * cos^2(\theta) [1 - \frac{v^2 }{c^2} ]+ L_o^2 * sin(\theta)^2}[/tex]
=> [tex]L_r = \sqrt{ [cos^2\theta +sin^2\theta ]- \frac{v^2 }{c^2}cos^2 \theta }[/tex]
=> [tex]L_o \sqrt{1 - \frac{v^2}{c^2 } cos^2(\theta ) }[/tex]
Hence the length of the rod as measured by a stationary observer is
[tex] L_r = L_o \sqrt{1 - \frac{v^2}{c^2 } cos^2(\theta ) }[/tex]
Generally the angle made is mathematically represented
[tex]tan(\theta) = \frac{L_y}{L_x}[/tex]
=> [tex]tan {\theta } = \frac{L_o sin(\theta )}{ (L_o cos(\theta ))\sqrt{ 1 -\frac{v^2}{c^2} } }[/tex]
=> [tex]tan(\theta ) = \frac{tan\theta}{\sqrt{1 - \frac{v^2}{c^2} } }[/tex]
Explanation:
The special relativity relations allow to find the results for the questions about the measurements made by an observed at rest on the rod are:
a) The length of the rod is: [tex]L = L_o \sqrt{1 - \frac{v^2}{c^2} \ cos^2\theta_o }[/tex]
b) The angle with respect to the x axis is: [tex]tan \theta = \frac{tan \theta_o}{\sqrt{1- \frac{v^2}{c^2} } }[/tex]
Special relativity studies the motion of bodies with speeds close to the speed of light, with two fundamental assumptions.
- The laws of physics are the same in all inertial systems.
- The speed of light in vacuum has the same value for all inertial systems.
If we assume that the two systems move in the x-axis, the relationship between the components of the length are:
[tex]L_x = L_{ox} \ \sqrt{1- \frac{v^2}{c^2} }[/tex]
[tex]L_y = L_o_y \\L_z = L_{oz}[/tex]
Where the subscript "o" is used for the fixed observed on the rod, that is, it is at rest with respect to the body, v and c are the speed of the system and light, respectively.
a) They indicate that the length of the rod is L₀ and it forms an angle θ with the horizontal.
Let's use trigonometry to find the components of the length of the rod in the system at rest, with respect to it.
sin θ = [tex]\frac{L_{oy}}{L_o}[/tex]
cos θ = [tex]\frac{L_{ox}}{L_o}[/tex]
[tex]L_{oy}[/tex] = L₀ sin θ
L₀ₓ = L₀ cos θ
Let us use the transformation relations of the length of the special relativity rod.
x-axis
[tex]L_x = (L_o cos \theta_o) \ \sqrt{1- \frac{v^2}{c^2} }[/tex]
y-axis
[tex]L_y = L_{o} sin \theta_o[/tex]
The length of the rod with respect to the observer using the Pythagorean theorem is:
L² = [tex]L_x^2 + L_y^2[/tex]
[tex]L^2 = (L_o cos \theta_o\sqrt{1- \frac{v^2}{c^2} })^2 + (L_o sin \theta_o)^2[/tex]
[tex]L_2 = L_o^2 ( cos^2 \theta_o - cos^2 \theta_o \frac{v^2}{c^2} + sin^2\theta_o)[/tex]
[tex]L^2 = L_o^2 ( 1 - \frac{v^2}{c^2} \ cos^2 \theta_o)[/tex]
[tex]L= Lo \sqrt{1- \frac{v^2}{c^2} cos^2 \theta_o}[/tex]
b) the angle with the x-axis measured by the stationary observer is:
[tex]tna \theta = \frac{L_y}{L_x}[/tex]
[tex]tan \ theta = \frac{L_o sin \theta_o}{L_o cos \theta_o \sqrt{1- \frac{v^2}{c^2} } }[/tex]
[tex]tan \theta = \frac{tan \theta_o}{\sqrt{1-\frac{v^2}{c^2} } }[/tex]
In conclusion, using the special relativity relations we can find the results for the questions about the measurements made by an observed at rest on the rod are:
a) The length of the rod is: [tex]L = L_o \sqrt{1- \frac{v^2}{c^2} \ cos^2\theta_o }[/tex]
b) The angle to the x axis is: [tex]tan \theta = \frac{tan \theta_o}{\sqrt{1- \frac{v^2}{c^2} } }[/tex]
Learn more about special relativity here: brainly.com/question/9820962