Question

A plank. fixed ro a sled at rest in frame \(S\), is of length \({L}_{0}\) and makes an angle of \(L_{0}\) with the \(x\) -axis. Later. the sled zooms through frame \(S\) at constant speed \(v\) parallel to the r-axis. Show that according to an observer who remains at rest in frame \(S\). the length of the plank is now\(L=L_{0} \sqrt{1-\frac{v^{2}}{c^{2}} \cos \theta_{0}}\) and the angle it makes with the \(x\) -axis is $$ \theta=\tan ^{-1}\left(\gamma_{v} \tan \theta_{0}\right) $$.

Short answer

The length and the angle of the plank as perceived by the observer in frame \(S\), after the sled starts moving are given by \(L = L_{0} \sqrt{1-\frac{v^{2}}{c^{2}} cos^2 \theta_0}\) and \(\theta = \tan^{-1}(\gamma_v \tan \theta_{0})\) respectively.

Step-by-step solution

Identify the Given Information

Initial parameters are given as follows: the plank's resting length is \(L_0\), the speed of the sled is \(v\), and the angle that the plank makes with the x-axis is \(\theta_0\). The task is to find the length of the plank and the angle it makes with the x-axis from the perspective of an observer in frame \(S\).

Apply the Formula for Length Contraction

As per the principle of length contraction in special relativity, the length of the plank as measured by the observer in stationary frame \(S\) after the sled starts moving is given by the formula \(L = L_{0} \sqrt{1-\frac{v^{2}}{c^{2}} cos^2 \theta_0}\). This formula accounts for the fact that length contraction happens only along the direction of motion (\(cos^2 \theta_0\) part).

Determine the Angle

Next, we need to calculate the angle the plank makes with the x-axis after the sled starts moving. This is given by \(\theta = \tan^{-1}(\gamma_v \tan \theta_{0})\), where \(\gamma_v = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\) is the Lorentz factor.

Key concepts

Special Relativity

Special relativity is a theory proposed by Albert Einstein that revolutionized our understanding of space and time. One of its key ideas is that the laws of physics are the same for all observers, regardless of their relative motion. This means, among other things, that neither time nor space are absolute. Instead, they are interconnected in what we call spacetime.

A fascinating consequence of special relativity is the phenomenon known as length contraction. When an object moves at a high velocity compared to the speed of light, it appears shorter in the direction of motion to a stationary observer. This effect becomes significant as the object approaches the speed of light.

For example, if a sled with a plank is moving quickly, an observer at rest would perceive the plank's length differently than an observer moving with the sled. This can be seen through special relativity equations, which show how significantly the length shortens based on speed and direction.

Lorentz Factor

The Lorentz factor, often denoted as \( \gamma \), is a crucial component of special relativity. It quantifies the amount of change in time and space dimensions for an object moving at relativistic speeds.

The Lorentz factor is defined as:\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]where \( v \) is the velocity of the moving object, and \( c \) is the speed of light.

• As an object’s speed \( v \) approaches the speed of light \( c \), \( \gamma \) increases dramatically. This means greater time dilation and length contraction occur.
• If \( v = 0 \), \( \gamma \) equals 1, signifying no relativistic effects.

In our scenario, the plank on the sled experiences these changes due to its motion. Consequently, the observer at rest uses the Lorentz factor to calculate the modified angle of the plank concerning the x-axis. The angle transformation is key, as the Lorentz factor affects how angles are perceived in a moving frame.

Angle Transformation

Angle transformation in the context of special relativity refers to how angles change when observed from different frames of reference. When analyzing the length of a moving plank, not only does its length contract, but the angle it makes with an axis transforms as well.

In our example, the original angle of the plank \( \theta_0 \) changes when the sled starts moving. For an observer at rest, this new angle \( \theta \) can be found using the formula:\[\theta = \tan^{-1}(\gamma_v \tan \theta_0)\]where \( \gamma_v \) is the Lorentz factor.

• This transformation shows that the angle depends on both the original angle and the velocity of the sled.
• As the speed increases and \( \gamma_v \) becomes larger, even small initial angles \( \theta_0 \) can lead to significant perceived changes.

Understanding angle transformation enriches our comprehension of how velocity affects not just an object's size, but also its orientation as observed from different frames.