Question

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

Short answer

Answer: None of the given statements are incorrect. All the statements (a), (b), (c), and (d) are true based on our analysis.

Step-by-step solution

Analysis of the statements and understanding the symbols

a) \(\beta=\sqrt{3} / 2\) Here, \(\beta\) represents the ratio of the speed of the object to the speed of light, i.e., \(\beta = v/c\). b) \(y=2\) \(y\) represents the Lorentz factor which is given by the formula \(y = \frac{1}{\sqrt{1-\beta^2}}\) c) To an observer at rest, it looks like a square with an area less than \(100 \mathrm{~m}^{2}\). We will analyze this statement by comparing the area of the square in different reference frames. d) The length along the direction of motion is contracted by a factor of \(\frac{1}{2}.\) This statement is about length contraction, and we will assess its correctness by comparing lengths contracted by a factor of \(\frac{1}{2}\).

Evaluating statement (a)

First, we check whether given \(\beta = \frac{\sqrt{3}}{2}\) is true. Given \(v = \frac{\sqrt{3}}{2}c\), the value of \(\beta\) the ratio of the speed of the object to the speed of light, is: \(\beta = \frac{v}{c} = \frac{\frac{\sqrt{3}}{2}c}{c} = \frac{\sqrt{3}}{2}\) Therefore, statement (a) is true.

Evaluating statement (b)

We will now check statement (b) by calculating the Lorentz factor \(y\) for the given \(\beta\): \(y = \frac{1}{\sqrt{1-\beta^2}} = \frac{1}{\sqrt{1 - (\frac{\sqrt{3}}{2})^2}} = \frac{1}{\sqrt{1 - \frac{3}{4}}} = \frac{1}{\sqrt{\frac{1}{4}}}=2\) Statement (b) is true.

Evaluating statement (c)

Now, let's determine if the area of the square seems smaller to an observer at rest. Due to length contraction, the length of the square along the direction of motion will be contracted, while the length perpendicular to the direction of motion remains unchanged. Let's assume the side length of the square in the rest frame is \(l\). Thus, the area of the square is \(100 \mathrm{~m}^{2} = l^2\) and \(l = 10\mathrm{m}\). In the moving frame, the side length parallel to the direction of motion will be contracted by a factor of \(y=2\). So, the contracted length is \(l' = l/2 = 5\mathrm{m}\). The side length perpendicular to the direction of motion remains unchanged at \(10\mathrm{m}\). The new area in the moving frame will be: \(A' = l' \times l = 5\mathrm{m} \times 10\mathrm{m} = 50 \mathrm{~m}^{2}\) Area in the moving frame is less than the area in the rest frame. Thus, statement (c) is true.

Evaluating statement (d)

Finally, let's analyze statement (d). The length contraction factor is given by \(\frac{1}{y} = \frac{1}{2}\), according to the problem information. This means that the length along the direction of motion is contracted by half. Comparing to our calculation in Step 4, where we found the contracted length to be \(l' = 5\mathrm{m}\), which is half of the original length \(l=10\mathrm{m}\), we can see that statement (d) is indeed true. Conclusion: Since statements (a), (b), (c), and (d) are all true, none of the given options is incorrect.