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.

Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

문제 27

35장, 문제 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 $\vec{v}$ = 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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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'.