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Ch. 36 - The Special Theory of Relativity
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 36 - The Special Theory of RelativityProblem 27
Chapter 35, Problem 27

A stick of length ℓ₀, at rest in reference frame S, makes an angle θ with the x axis. In reference frame S', which moves to the right with velocity v→\(\overrightarrow{v}\) = vî with respect to S, determine (a) the length l of the stick, and (b) the angle θ it makes with the x' axis.

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To solve part (a), recall the concept of length contraction in special relativity. The length of an object parallel to the direction of motion is contracted by a factor of γ = 1 / √(1 - v²/c²), where v is the relative velocity and c is the speed of light. The component of the stick's length along the x-axis, ℓₓ = ℓ₀ cos(θ), will be contracted in the S' frame, while the component perpendicular to the x-axis, ℓᵧ = ℓ₀ sin(θ), remains unchanged.
The contracted length along the x-axis in the S' frame is given by ℓₓ' = ℓₓ / γ = (ℓ₀ cos(θ)) / √(1 - v²/c²). The total length of the stick in the S' frame, ℓ', can then be calculated using the Pythagorean theorem: ℓ' = √(ℓₓ'² + ℓᵧ²). Substitute ℓₓ' and ℓᵧ into this equation to express ℓ' in terms of ℓ₀, θ, v, and c.
For part (b), the angle θ' that the stick makes with the x'-axis in the S' frame can be determined using trigonometry. The tangent of the angle is given by tan(θ') = ℓᵧ / ℓₓ'. Substitute ℓᵧ = ℓ₀ sin(θ) and ℓₓ' = (ℓ₀ cos(θ)) / √(1 - v²/c²) into this expression to find tan(θ').
Simplify the expression for tan(θ') to obtain θ' in terms of θ, v, and c. Use the relationship tan(θ') = sin(θ') / cos(θ') to express θ' explicitly if needed.
Finally, summarize the results: (a) the contracted length ℓ' is determined by the Pythagorean theorem using the contracted and unchanged components, and (b) the angle θ' is found using the tangent relationship. These results demonstrate how relativistic effects alter both the length and orientation of the stick in the moving reference frame.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Length Contraction

Length contraction is a phenomenon predicted by the theory of special relativity, where the length of an object moving relative to an observer is measured to be shorter than its proper length (the length measured in the object's rest frame). This effect becomes significant at velocities approaching the speed of light and is given by the formula L = L₀√(1 - v²/c²), where L₀ is the proper length, v is the relative velocity, and c is the speed of light.
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Reference Frames

A reference frame is a perspective from which measurements are made, including the position and motion of objects. In this problem, S is the stationary frame where the stick is at rest, while S' is the moving frame where the stick's length and angle are observed differently due to its motion. Understanding how different reference frames affect measurements is crucial in relativity.
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Relativistic Angles

Relativistic angles refer to how the angle of an object changes when observed from different reference frames, particularly when one frame is moving relative to another. The transformation of angles involves not only the effects of length contraction but also the Lorentz transformation, which relates the coordinates of events in different frames. This concept is essential for determining how the angle θ of the stick changes when viewed from frame S'.
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Related Practice
Textbook Question

An observer in reference frame S notes that two events are separated in space by 180 m and in time by 0.80μs. How fast must reference frame S' be moving relative to S in order for an observer in S' to detect the two events as occurring at the same location in space?

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Textbook Question

(III) If a particle moves in the xy plane of system S (Fig. 36–12) with speed u in a direction that makes an angle θ with the x axis, show that it makes an angle θ' in S' given by tanθ′=(sinθ)1−v2/c2/(cosθ−v/u)\(\tan\]\theta\)^{\(\prime\)}=(\(\sin\[\theta\))\(\sqrt{1-v^2/c^2}\)/(\(\cos\]\theta\)-v/u).

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Textbook Question

In the old West, a marshal riding on a train traveling 35.0 m/s sees a duel between two men standing on the Earth 55.0 m apart parallel to the train. The marshal’s instruments indicate that in his reference frame the two men fired simultaneously.

(a) Which of the two men, the first one the train passes (A) or the second one (B) should be arrested for firing the first shot? That is, in the gunfighter’s frame of reference, who fired first?

(b) How much earlier did he fire?

(c) Who was struck first?

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Textbook Question

A spaceship traveling at 0.76c away from Earth fires a module with a speed of 0.85c at right angles to its own direction of travel (as seen by the spaceship). What is the speed of the module, and its direction of travel (relative to the spaceship’s direction), seen by an observer on Earth?

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Textbook Question

Two spaceships leave Earth in opposite directions, each with a speed of 0.50c with respect to Earth.

(a) What is the velocity of spaceship 1 relative to spaceship 2?

(b) What is the velocity of spaceship 2 relative to spaceship 1?

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Textbook Question

(I) Repeat Problem 19 using the Lorentz transformation and a relative speed v = 1.60 x 10⁸ m/s, but choose the time t to be (a) 3.5μs and (b) 10.0 μs .

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