Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

A cube of metal with sides of length \(a\) sits at rest in a frame \(S\) with one edge parallel to the \(x\)-axis. Therefore, in \(S\) the cube has volume \(a^3\). Frame \(S'\) moves along the \(x\)-axis with a speed \(u\). As measured by an observer in frame \(S'\), what is the volume of the metal cube?

Short Answer

Expert verified
The volume is \(a^3 \sqrt{1 - \frac{u^2}{c^2}}\)."

Step by step solution

01

Understanding Relativity's Effect on Dimensions

When an object moves at a significant fraction of the speed of light, its length along the direction of motion contracts due to the phenomenon known as length contraction. This contraction does not affect dimensions perpendicular to the direction of motion. Here, the cube is moving along the x-axis in frame \(S'\), so only the length along the x-axis is affected.
02

Apply Length Contraction Formula

The length contraction formula is \(L' = L \sqrt{1 - \frac{u^2}{c^2}}\), where \(L'\) is the contracted length, \(L\) is the rest length, \(u\) is the relative velocity, and \(c\) is the speed of light. Since the cube's side \(a\) is parallel to the x-axis, it becomes \(a' = a \sqrt{1 - \frac{u^2}{c^2}}\).
03

Calculate the Transverse Dimensions

Because the movement does not affect the other dimensions of the cube transverse to the direction of motion, the sides parallel to the y and z axes remain the same, \(a\).
04

Calculate the Volume in Frame S'

The volume of the cube in frame \(S'\) is given by the product of its dimensions: \(V' = a' \cdot a \cdot a = (a \sqrt{1 - \frac{u^2}{c^2}}) \cdot a \cdot a = a^3 \sqrt{1 - \frac{u^2}{c^2}}\). This is the volume as observed in frame \(S'\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Special Relativity
Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It provided a new way to understand the relationship between space and time, drastically changing how we perceive motion. The core idea is that the laws of physics are the same for all observers, regardless of their relative motion. One of the groundbreaking concepts introduced by special relativity is that the speed of light in a vacuum is constant and will be the same for all observers, regardless of their motion relative to the light source.
This invariance of the speed of light leads to some counterintuitive consequences. For example, time dilation, where a moving clock ticks slower than one at rest, and length contraction, which we will delve into more in upcoming sections. The theory of special relativity highlights how measurements of time and space are not absolute but depend on the observer's state of motion. This revelation paved the way for modern physics to explain phenomena occurring at very high speeds, close to the speed of light.
Lorentz Transformation
The Lorentz transformation equations are mathematical formulas used to switch between two reference frames that are moving at a constant velocity relative to each other. These transformations explain how the coordinates of time and space are related in different inertial frames. Named after the physicist Hendrik Lorentz, these equations incorporate the factor \( \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \), where \(c\) is the speed of light and \(u\) is the relative velocity.
The main use of the Lorentz transformation is to ensure that the laws of physics, especially those involving electromagnetic phenomena, remain consistent for different observers, regardless of their motion. In particular, they account for changes in time, length, and simultaneity that occur when objects move at velocities approaching the speed of light. When applied to the case of length contraction, these transformations show that an object moving in regard to an observer will appear shorter along the direction of motion by the factor of \(\sqrt{1 - \frac{u^2}{c^2}}\). This is a crucial part in understanding how and why the volume of objects changes in different inertial frames.
Relativistic Effects
Relativistic effects become noticeable when objects move at speeds close to the speed of light. These include time dilation and length contraction, both of which are direct consequences of special relativity. Length contraction, the focus of our original exercise, occurs because the length of an object along the direction of motion decreases as its speed approaches the speed of light. This effect only impacts dimensions parallel to the direction of travel. For instance, in the scenario of the metal cube, only the side aligned with the motion direction is contracted. By contrast, dimensions perpendicular to the motion remain unchanged.
These relativistic effects are not just theoretical concepts. They have practical implications in modern technology and experiments. For example, the Global Positioning System (GPS) accounts for these effects to provide accurate positioning data, as the satellites move at high speeds relative to the Earth. Understanding these effects is essential in fields like astrophysics and particle physics, where high-velocity particles are more commonly encountered. Thus, mastering the concept of length contraction and other relativistic effects is crucial for comprehending the universe's behavior at its most fundamental levels.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth's surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of 0.920c relative to the spaceship. An observer on Arrakis measures that the rocket is approaching with a speed of 0.360c. What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

(a) How much work must be done on a particle with mass \(m\) to accelerate it (a) from rest to a speed of 0.090\(c\) and (b) from a speed of 0.900\(c\) to a speed of 0.990\(c\) ? (Express the answers in terms of \(mc^2\).) (c) How do your answers in parts (a) and (b) compare?

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed \(V\) relative to the lab frame is $$v = {c \over n} + kV$$ where \(n\) = 1.333 is the index of refraction of water. Fizeau called \(k\) the dragging coefficient and obtained an experimental value of \(k\) = 0.44. What value of \(k\) do you calculate from relativistic transformations?

In the earth's rest frame, two protons are moving away from each other at equal speed. In the frame of each proton, the other proton has a speed of 0.700\(c\). What does an observer in the rest frame of the earth measure for the speed of each proton?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free