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Chapter 18: Problem 19

Imagine two protoms traveling past each other at a distance \(d,\) with relative speed \(0.9 c .\) Compared with two stationary protons a distance \(d\) apart, the gravitational force between these two protons will be a. smaller, because they interact for less time. b. smaller, because the moving proton acts as if it has less mass. c. the same, because in both cases, the mass of the two particles are identical. d. larger, because the moving proton acts as if it has more mass.

Short Answer

Expert verified
The gravitational force between the moving protons will be larger because the moving proton acts as if it has more mass due to relativistic effects. The correct answer is 'd.'

Step by step solution

01

- Analyze the Problem Statement

Understand the relationship and interaction between the two protons. The gravitational force between protons is usually determined by their masses and the distance between them.
02

- Recall the Gravitational Force Formula

The gravitational force between two protons is given by Newton's law of gravitation: \[ F = \frac{G m_1 m_2}{d^2} \]where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the protons, and \(d\) is the distance between them.
03

- Consider the Effect of Relativity on Mass

When an object moves at a significant fraction of the speed of light, its relativistic mass increases. According to special relativity, the relativistic mass is given by: \[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \(m_0\) is the rest mass, \(v\) is the velocity of the object, and \(c\) is the speed of light. For the given problem, \( v = 0.9c\).
04

- Calculate Relativistic Mass

Calculate the increase in mass due to the high velocity:\[m = \frac{m_0}{\sqrt{1 - (0.9c)^2 / c^2}} = \frac{m_0}{\sqrt{1 - 0.81}} = \frac{m_0}{\sqrt{0.19}} \]This means that the moving proton appears to have more mass compared to a stationary proton.
05

- Effect of Increased Mass on Gravitational Force

Since the mass appears larger, the gravitational force is affected. Inserting the relativistic mass into the gravitational formula shows that the force will indeed be larger.
06

- Verify the Correct Option

Based on the calculation, the moving proton acts as if it has more mass. This leads to a larger gravitational force between them. Hence, the correct answer is option 'd'.

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Key Concepts

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

Relativistic Mass
When an object moves at speeds close to the speed of light, its mass increases. This phenomenon is called relativistic mass. According to special relativity, the mass of an object moving at velocity \(v\) is calculated as:
\[ m = \frac{m_0}{\text{sqrt}(1 - \frac{v^2}{c^2})} \]
Here:
  • \( m_0 \) is the rest mass.
  • \( v \) is the object's velocity.
  • \( c \) is the speed of light.
For protons moving at \(0.9c\), their mass increases significantly. This adjustment in mass affects other calculations, like the gravitational force between them.
Newton's Law of Gravitation
Newton's law of gravitation explains the gravitational force between two bodies. The equation is:
\[ F = \frac{G m_1 m_2}{d^2} \]
In this equation:
  • \( F \) is the gravitational force.
  • \( G \) is the gravitational constant.
  • \( m_1 \) and \( m_2 \) are the masses of the two bodies.
  • \( d \) is the distance between them.
For protons, using their relativistic mass instead of their rest mass will change the gravitational force due to their increased mass.
Special Relativity
Special relativity, proposed by Albert Einstein, has profound implications on how we understand space and time. One of its key principles is that an object's mass increases as its speed approaches the speed of light. This can be seen in our problem, where protons traveling at \(0.9c\) have their mass significantly increased. This also implies:
  • Time dilation: Moving clocks run slower compared to stationary clocks.
  • Length contraction: Moving objects are measured to be shorter along the direction of motion.
These effects become noticeable at velocities close to the speed of light.
Gravitational Constant
The gravitational constant (\(G\)) is a fundamental constant crucial in Newton's law of gravitation. It is approximately equal to \[6.674 \times 10^{-11} \frac{\text{m}^3}{\text{kg} \times \text{s}^2} \]
This value helps determine the force of gravity between two objects based on their masses and the distance between them. In our problem, the gravitational force between two protons is directly proportional to the product of their relativistic masses and inversely proportional to the square of the distance.
Proton Interactions
Protons, which are positively charged particles, do not naturally attract each other by electric force because like charges repel. However, they do interact via gravitational force, although it is significantly weaker compared to other fundamental forces. In our exercise, the gravitational force between two protons, moving at high speeds, increases due to the increase in their relativistic mass. This interaction demonstrates how combining concepts from special relativity and Newtonian gravity can impact our understanding of particle physics.

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