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Chapter 37: Problem 38

A rod is to move at constant speed \(v\) along the \(x\) axis of reference frame \(S\), with the rod's length parallel to that axis. An observer in frame \(S\) is to measure the length \(L\) of the rod. Figure 37-17 gives length \(L\) versus speed parameter \(\beta\) for a range of values for \(\beta\). The vertical axis scale is set by \(L_{a}=1.00 \mathrm{~m}\). What is \(L\) if \(v=0.90 c\) ?

Short Answer

Expert verified
The length \(L\) is approximately 0.44 m.

Step by step solution

01

Understanding the Problem

We need to find the length of the rod as observed in reference frame \(S\) when the rod's speed \(v\) is \(0.90c\), where \(c\) is the speed of light. The length of the rod changes according to the formula involving the Lorentz factor.
02

Correct Formula

The length contraction formula is used, which is given by \( L = L_0 \sqrt{1 - \beta^2} \), where \( L_0 \) is the proper length of the rod and \( \beta = \frac{v}{c} \).
03

Substitute Given Values

Given \( \beta = 0.90 \), we substitute to find \(L\). So, \(L = L_a \sqrt{1 - (0.90)^2} \). We know \(L_a = 1.00 \, \text{m}\).
04

Simplify the Expression

Calculate the expression. First, find \((0.90)^2 = 0.81\). Then, \(1 - 0.81 = 0.19\). Thus, \(L = 1.00 \, \text{m} \times \sqrt{0.19}\).
05

Compute Final Length

Evaluate \(\sqrt{0.19} \approx 0.4359\). Therefore, \(L = 1.00 \, \text{m} \times 0.4359 \approx 0.4359 \, \text{m}\).

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Key Concepts

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

Relativity
Albert Einstein's theory of relativity is an impactful theory in physics that mainly includes two principles: the special and general theories. For this task, we focus on special relativity, which deals with objects moving at high speeds, similar to the speed of light.
This theory introduce a new way to think about how time and space are linked and how they behave differently for observers in different states of motion.
For example, length contraction is a portion of special relativity that implies that the measured length of an object in motion will be shorter than its actual length when measured at rest.
Lorentz Factor
The Lorentz factor, often denoted by the Greek letter \( \gamma \), shows how much time, length, and relativistic effects change by high-speed motion.
It is defined by the equation \( \gamma = \frac{1}{\sqrt{1-\beta^2}} \), where \( \beta = \frac{v}{c} \) and \( v \) is the velocity of the object, while \( c \) is the speed of light.
The Lorentz factor can become quite large when an object鈥檚 speed approaches the speed of light, indicating more visible relativistic effects like time dilation and length contraction.
  • Time dilation: Time moves slower for a fast-moving observer compared to one at rest.
  • Length contraction: Moving objects appear shorter along the direction of motion.
These effects become especially pronounced as \( v \) approaches \( c \).
Proper Length
The proper length refers to the length of an object measured in the object's rest frame. This means the frame where it's not moving.
The proper length \( L_0 \) is an object's longest possible measured length, and any observer measuring the object in any other frame of reference would measure a shorter length due to relativistic effects.
Specifically, for the concept of length contraction, this proper length is essential. Observers watching the same object but from different moving frames will measure various lengths, but they all base on the object's proper length.
Reference Frame
A reference frame is essentially a perspective from which you鈥檙e observing or measuring physical phenomena. These frames are crucial for comparing observations made in different scenarios, especially under the influence of relativity.
There are two types of reference frames 鈥�
  • **Inertial reference frames:** where objects move at a constant velocity, not accelerating.
  • **Non-inertial reference frames:** where observers experience acceleration.
In our exercise, frame \(S\) is an inertial frame with the observer being in this particular reference frame.
Understanding reference frames facilitates grasping why different observers might measure different times, lengths, or speeds of the same event or object.

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Most popular questions from this chapter

A spaceship is moving away from Earth at speed \(0.333 c\). A source on the rear of the ship emits light at wavelength \(450 \mathrm{~nm}\) according to someone on the ship. What (a) wavelength and (b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?

Inertial frame \(S^{\prime}\) moves at a speed of \(0.65 c\) with respect to frame \(S\) (Fig. 37-9). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime}=0\). Two events are recorded. In frame \(S\), event 1 occurs at the origin at \(t=0\) and event 2 occurs on the \(x\) axis at \(x=3.0 \mathrm{~km}\) at \(t=4.0 \mu\) s. According to observer \(S^{\prime}\), what is the time of (a) event 1 and (b) event \(2 ?\) (c) Do the two observers see the same sequence or the reverse?

Galaxy \(A\) is reported to be receding from us with a speed of \(0.45 c\). Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What multiple of \(c\) gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?

What must be the momentum of a particle with mass \(m\) so that the total energy of the particle is \(4.00\) times its rest energy?

The center of our Milky Way galaxy is about 23000 ly away. (a) To eight significant figures, at what constant speed parameter would you need to travel exactly 23000 ly (measured in the Galaxy frame) in exactly \(40 \mathrm{y}\) (measured in your frame)? (b) Measured in your frame and in light- years, what length of the Galaxy would pass by you during the trip?
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