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Chapter 2: Problem 15

A famous relation in physics relates 'moving mass' \(m\) to the 'rest mass' \(m_{0}\) of a particle in terms of its speed \(v\) and the speed of light, \(c\). (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes: \(m=\frac{m_{0}}{\left(1-v^{2}\right)^{1 / 2}}\) Guess where to put the missing \(c\).

Short Answer

Expert verified
The missing \(c\) should be in the denominator as \(c^2\) under the square root: \(m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\).

Step by step solution

01

Understanding the Context

The exercise involves a relation from Einstein's theory of special relativity, which involves the moving mass of a particle in relation to its rest mass and its velocity compared to the speed of light.
02

Analyzing the Given Equation

The boy's equation is given as: \[ m = \frac{m_0}{\sqrt{1-v^2}} \]It does not include the speed of light constant \(c\), which is crucial in special relativity.
03

Identifying the Role of Speed of Light

In the context of special relativity, the speed of light \(c\) is a fundamental constant that should be included in the denominator of the equation to correctly relate the mass and speed of a particle.
04

Correcting the Equation

The correct formula according to special relativity is: \[ m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \]In this equation, \(c^2\) is placed in the denominator within the square root to ensure that the equation is dimensionally correct.

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Key Concepts

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

Moving Mass
In the fascinating world of special relativity, the concept of 'moving mass' is crucial when examining objects that travel at significant fractions of the speed of light. When an object moves, its mass appears increased from the perspective of an outside observer. This is referred to as the 'moving mass', and it highlights how mass can depend on velocity in the relativistic framework.
  • Moving mass is described by the equation: \[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] Here, the variable \(m\) refers to the moving mass, and \(v\) is the velocity of the object.
  • The increased mass influences the object's inertia and energy as it travels closer to the speed of light. The heavier mass requires more force to accelerate, in accordance with Newton鈥檚 second law.
The moving mass becomes significant in high-energy physics, where particles approach the speed of light, illustrating the deep interplay between energy, mass, and velocity.
Rest Mass
To comprehend moving mass, we must first understand 'rest mass' (\(m_0\)). Rest mass is the intrinsic mass of an object when it is at rest, effectively a measure of how much matter it contains disengaged from any relative velocity effects.
  • This concept is fundamental in special relativity as it remains constant, unaffected by an object's motion or orientation, unlike moving mass.
  • Rest mass acts as a baseline before accounting for the transformative effects of motion on mass, which arise distinctly at relativistic speeds.
The formula for moving mass incorporates rest mass, highlighting the transition and relationship between motion and mass that are central to Einstein's theory.
Speed of Light
The speed of light, denoted by \(c\), is one of the universe's fundamental constants. It plays a pivotal role in the equations of special relativity and profoundly affects how velocities and times are measured at high speeds.
  • Speed of light in vacuum is approximately \(299,792,458\) meters per second. This immense speed acts as the ultimate speed limit for the transfer of information and matter.
  • In special relativity, \(c\) ensures that equations remain consistent regardless of relative motion, underpinning the universal principle that laws of physics are the same for all observers.
Its consistent role in relativistic equations, like \(m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}\), ensures that as a particle's speed approaches \(c\), its mass tends towards infinity, making further acceleration impossible.
Einstein's Theory
Albert Einstein's groundbreaking theory of special relativity radically altered our understanding of space, time, and mass.
  • Prior to Einstein, mass and time were viewed as absolute. Einstein demonstrated that they are relative by introducing new concepts like time dilation and length contraction.
  • Special relativity's core propositions rely on two postulates:
    • The laws of physics are the same in all inertial frames, meaning no preferred inertial vantage point exists.
    • The speed of light in a vacuum is constant across all frames of reference.
These ideas reshaped physics by suggesting that observers moving at different velocities will have distinct experiences of time and space, reshaping the way we understand our cosmos. The relation between moving and rest mass is just one way special relativity weaves a comprehensive framework for understanding high-speed physical phenomena.

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Most popular questions from this chapter

Explain this common observation clearly: If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to the train's motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these wi

Fill in the blanks by suitable conversion of units (a) \(1 \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-2}=\ldots . \mathrm{g} \mathrm{cm}^{2} \mathrm{~s}^{-2}\) (b) \(1 \mathrm{~m}=\ldots . \mathrm{ly}\) (c) \(3.0 \mathrm{~m} \mathrm{~s}^{2}=\ldots . \mathrm{km} \mathrm{h}^{2}\) (d) \(G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2}(\mathrm{~kg})^{-2}=\ldots .\left(\mathrm{cml}^{3} \mathrm{~s}^{-2} \mathrm{~g}^{-1} .\right.\)

The unit of length convenient on the atomic scale is known as an angstrom and is denoted by \(\AA\) : \(1 \AA^{\circ}=10^{-10} \mathrm{~m}\). The size of a hydrogen atom is about \(0.5 \AA\). What is the total atomic volume in \(\mathrm{m}^{3}\) of a mole of hydrogen atoms?

State the number of significant figures in the following: (a) \(0.007 \mathrm{~m}^{2}\) (b) \(2.64 \times 10^{24} \mathrm{~kg}\) (c) \(0.2370 \mathrm{~g} \mathrm{~cm}^{-3}\) (d) \(6.320 \mathrm{~J}\) (e) \(6.032 \mathrm{~N} \mathrm{~m}^{-2}\) (f) \(0.0006032 \mathrm{~m}^{2}\)

It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about \(0.02 \mathrm{~s}\). What does this imply for the accuracy of the standard cesium clock in measuring a time- interval of \(1 \mathrm{~s} ?\)
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