Question

A square of area \(100 \mathrm{~m}^{2}\) that is at rest in the reference frame is moving with a speed \((\sqrt{3} / 2) c\). Which of the following statements is incorrect? a) \(\beta=\sqrt{3} / 2\) b) \(\gamma=2\) c) To an observer at rest, it looks like another square with an area less than \(100 \mathrm{~m}^{2}\) d) The length along the moving direction is contracted by a factor of \(\frac{1}{2}\)

Short answer

Based on the analysis and calculations performed, all four statements, a), b), c), and d) are correct. Therefore, there is no incorrect statement among the given options, indicating an error in the problem itself.

Step-by-step solution

Determine \(\beta\)

The first statement says that \(\beta = \sqrt{3} / 2\). We know that \(\beta = \frac{v}{c}\), so let's check if this statement is correct: \[\beta = \frac{v}{c} = \frac{\frac{\sqrt{3}}{{~}}c}{c} = \frac{\sqrt{3}}{2}\] So, statement a) is correct.

Determine \(\gamma\)

The second statement says that \(\gamma = 2\). The Lorentz factor is given by \(\gamma = \frac{1}{\sqrt{1 - \beta^2}}\). Using the value of \(\beta\) we have just calculated: \[\gamma = \frac{1}{\sqrt{1 - \left(\frac{\sqrt{3}}{2}\right)^2}} = \frac{1}{\sqrt{1 - \frac{3}{4}}} = \frac{1}{\sqrt{\frac{1}{4}}} = 2\] So, statement b) is correct.

Analyze Area under different reference frame

For an observer at rest, we need to find whether the area of the moving square is less than 100 m². The length contraction takes place only along the direction of motion, so we consider one side of the square and find the contracted length: \[L'_{x} = \frac{L_{x}}{\gamma} = \frac{\sqrt{100}}{2} = 5\sqrt{2}\] Since only one side is contracted, the area of the square in the rest frame of the observer will be: \[A' = L'_{x} * L_{y} = 5\sqrt{2} * 10 = 50\sqrt{2}\] As \(50\sqrt{2} < 100\), the statement c) is correct.

Check Length Contraction Factor

The fourth statement says that the length along the moving direction is contracted by a factor of \(\frac{1}{2}\). We have already calculated the contracted length \(L'_{x}\) and original length \(L_{x}\): \[\frac{L'_{x}}{L_{x}} = \frac{5\sqrt{2}}{10} = \frac{1}{2}\] So, statement d) is also correct. Since the problem asks for the incorrect statement and all four statements are correct, it seems there is an error in the problem itself. Each statement provided is accurate according to our calculations.

Key concepts

Length Contraction

Length contraction is a fascinating phenomenon from the theory of special relativity. It states that an object moving relative to an observer will appear shorter in the direction of its motion. This effect only becomes significant at velocities close to the speed of light.

In our exercise, a square moves at a high speed of \(\left(\frac{\sqrt{3}}{2}\right)c\), causing its observed length to contract.
The formula for length contraction is given by: \[L' = \frac{L}{\gamma}\] where \(L'\) is the contracted length, \(L\) is the proper length (length of the object in its rest frame), and \(\gamma\) is the Lorentz factor.

The contraction occurs only along the direction of motion. Hence, while one side of the square contracts, the perpendicular side remains unchanged. This leads to a change in the apparent area of the square from the observer's point of view.

Understanding length contraction helps us comprehend how size and distance can vary depending on an observer's frame of reference when dealing with relativistic speeds.

Lorentz Factor (γ)

The Lorentz factor, often denoted as \(\gamma\), is a crucial component in the equations of special relativity. It quantifies the various relativistic effects, such as time dilation and length contraction. The Lorentz factor is defined as: \[\gamma = \frac{1}{\sqrt{1- \beta^2}}\]where \(\beta = \frac{v}{c}\), with \(v\) being the velocity of the object and \(c\) being the speed of light.

In our exercise, \(\beta = \frac{\sqrt{3}}{2}\), which gives \(\gamma = 2\). This value shows that relativistic effects are pronounced at such high speeds.

• The Lorentz factor becomes significantly larger than 1 as speeds approach that of light.
• It makes the object's dimensions along the direction of motion contract (length contraction).
• It also affects time, causing it to dilate (slow down) relative to a stationary observer.

The Lorentz factor allows us to accurately describe the changes observed in physical measurements across different reference frames when dealing with relativistic speeds.

Relativistic Effects

Relativistic effects describe how the laws of physics change when dealing with objects moving at speeds close to that of light. These effects include: > **Length contraction**: As already discussed, it causes objects to shorten in the direction of motion, affecting both their dimensions and apparent shape from a stationary observer's standpoint.

> **Time dilation**: As speeds increase, time appears to move slower for the moving object relative to a stationary observer. This can lead to fascinating scenarios like the "twin paradox," where one twin ages slower while traveling at relativistic speeds.

> **Mass increase**: According to relativity, an object's mass increases with velocity, making it harder to accelerate as speed approaches the speed of light. These effects are not noticeable at everyday speeds but become crucial at velocities close to that of light. Einstein's theory of relativity provides the framework to understand and calculate these interactions, ensuring the consistency of the laws of physics across all inertial frames.