Question

(a) A meter stick is at rest in frame \(\mathcal{S}_{\mathrm{o}}\), which is traveling with speed \(V=0.8 c\) in the standard configuration relative to frame \(\mathcal{S}\). (a) The stick lies in the \(x_{\mathrm{o}} y_{\mathrm{o}}\) plane and makes an angle \(\theta_{\mathrm{o}}=60^{\circ}\) with the \(x_{\mathrm{o}}\) axis (as measured in \(\mathcal{S}_{\mathrm{o}}\) ). What is its length \(l\) as measured in \(\mathcal{S}\), and what is its angle \(\theta\) with the \(x\) axis? [Hint: It may help to think of the stick as the hypotenuse of a \(30-60-90\) triangle of plywood.] (b) What is \(l\) if \(\theta=60^{\circ} ?\) What is \(\theta_{\mathrm{o}}\) in this case?

Short answer

(a) Length \(l = 0.866\,\text{m}\), angle \(\theta \approx 79.1^\circ\). (b) \(l = 1\,\text{m}\), \(\theta_o = 60^\circ\).

Step-by-step solution

Understanding the problem

The problem involves a meter stick at rest in frame \(\mathcal{S}_o\) which is moving relative to another frame \(\mathcal{S}\) with a speed \(V = 0.8c\). The stick makes an angle of 60° with the \(x_o\) axis. We need to find its length \(l\) and angle \(\theta\) in frame \(\mathcal{S}\), and consider what happens when the stick's angle \(\theta\) is 60° in this moving frame.

Analyze the initial setup in \(\mathcal{S}_o\)

In frame \(\mathcal{S}_o\), the stick is 1 meter long and oriented at an angle \(\theta_o = 60^\circ\). Since it's a 30-60-90 triangle, the lengths of the perpendicular sides can be calculated using trigonometric functions: the side along the \(x_o\)-axis is \(\cos 60^\circ\) and along the \(y_o\)-axis is \(\sin 60^\circ\), meaning the lengths are \(0.5 \, \text{m}\) and \(\sqrt{3}/2 \, \text{m}\) respectively.

Apply Length contraction in \(\mathcal{S}\)

In \(\mathcal{S}\), the length of the stick along the direction of motion (\(x\)-axis) experiences relativistic length contraction. The contracted length \(L_x\) is given by:\[ L_x = L_{x_o} \sqrt{1 - \left(\frac{V}{c}\right)^2} = 0.5 \times \sqrt{1 - 0.8^2} = 0.5 \times 0.6 = 0.3 \, \text{m} \]The length along the \(y\)-axis remains unchanged as length contraction only affects dimensions parallel to the motion.

Calculate the Length and Angle in \(\mathcal{S}\)

With \(L_x = 0.3 \, \text{m}\) and \(L_y = \sqrt{3}/2 \, \text{m}\), the actual length \(l\) in frame \(\mathcal{S}\) is the hypotenuse of the right triangle:\[ l = \sqrt{L_x^2 + L_y^2} = \sqrt{0.3^2 + (\sqrt{3}/2)^2} \approx 0.866 \, \text{m} \]The angle \(\theta\) with the \(x\)-axis is given by:\[ \tan \theta = \frac{L_y}{L_x} = \frac{\sqrt{3}/2}{0.3} \]\[ \theta = \arctan \left(\frac{\sqrt{3}}{0.6}\right) \approx 79.1^\circ\]

Transform Relative Conditions

For Part (b), if the angle \(\theta = 60^\circ\) in frame \(\mathcal{S}\), solve for \(l\):Knowing \(\tan\theta = \frac{L_y}{L_x}\), for \(\theta = 60^\circ\), \(\tan 60^\circ = \sqrt{3}\), leading to:\[ \frac{L_y}{L_x} = \sqrt{3} \rightarrow L_y = \sqrt{3}L_x \]Thus, with \(L_x = L_{x_o} \times 0.6\), calculate \(l = \sqrt{L_x^2 + L_y^2}\):\[ l = \sqrt{(L_{x_o} \times 0.6)^2 + (\sqrt{3}L_{x_o} \times 0.6)^2} \approx 1 \, \text{m} \]To find \(\theta_o\), if \(L_y = \sqrt{3} \times L_x = \text{initial }L_y\), the initial angle \(\theta_o\) remains \(60^\circ\) but is dependent on relative perception.

Key concepts

Length Contraction

One of the intriguing concepts in the realm of special relativity is length contraction. This phenomenon occurs because of the nature of space and time as described by Einstein's theory. When an object, such as a meter stick, moves at a high velocity relative to an observer, its length appears shorter to that observer when measured along the direction of motion.

This contraction is quantified by the Lorentz factor, given by the formula:

• The contracted length, \(L'\), can be calculated using: \[ L' = L \sqrt{1 - \frac{V^2}{c^2}} \] where \(L\) is the original length in its rest frame, \(V\) is the velocity of the object, and \(c\) is the speed of light.
• The contraction only affects the dimension parallel to the velocity.

In our specific problem, this contraction is evident as the meter stick travels with speed \(0.8c\). Calculations lead us to find that the contracted length along the motion is just \(0.3 \text{ m}\), proving the concept as real and calculable under given conditions.

Reference Frames

Understanding reference frames is crucial when diving into special relativity. Frames of reference dictate how we perceive motion and position. In the textbook problem, we are dealing primarily with two frames: \(\mathcal{S}_o\) (the rest frame of the stick) and \(\mathcal{S}\) (the moving observer's frame).

Here are key components to grasp about reference frames:

• Each frame of reference can have different measurements for time and space due to their relative motion.
• The concept ensures that physical phenomena remain consistent, yet they may appear differently to observers in relative motion.
• Transforming between these frames requires us to apply the Lorentz transformation equations.

In the case at hand, the stick appears at different lengths and angles when viewed from these distinct frames of reference. It underscores how measurements are not absolute but rely on the observer's frame.

Lorentz Transformation

The Lorentz Transformation bridges two reference frames that are moving relative to one another. This transformation is fundamental in calculating how quantities like time, length, and angles change due to special relativity.

The equations involved in Lorentz transformations allow us to convert coordinates from one frame to another:

• The change in time and space is accounted by: \[ x' = \frac{x - Vt}{\sqrt{1 - \frac{V^2}{c^2}}} \] \[ t' = \frac{t - \frac{Vx}{c^2}}{\sqrt{1 - \frac{V^2}{c^2}}} \]
• For angles, like in our scenario, the transformation alters their perception:
• Since the stick's perceived length and angle change, \(\tan \theta\) can be derived using the transformed lengths.

Understanding such transformations is vital in predicting how objects in motion are observed differently across frames. This allows us, for example, to see why the length contraction mentioned for the stick matters under the conditions given.