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Chapter 37: Problem 71

In a particle accelerator a proton moves with constant speed 0.750\(c\) in a circle of radius 628 \(\mathrm{m} .\) What is the net force on the proton?

Short Answer

Expert verified
The net force on the proton is approximately \(1.35 \times 10^{-10} \text{ N}\).

Step by step solution

01

Identify Known Values

We know the speed of the proton is given as 0.750\(c\), where \(c\) is the speed of light \(c = 3.00 \times 10^8 \text{ m/s}\). The radius of the circle is \(r = 628 \text{ m}\).
02

Calculate Proton Speed

Calculate the speed of the proton using the constant speed factor. Speed \( v = 0.750 \times c = 0.750 \times 3.00 \times 10^8 = 2.25 \times 10^8 \text{ m/s}\).
03

Apply Centripetal Force Formula

The net force acting on the proton moving in a circular path is the centripetal force. This force is given by the formula \( F = \frac{mv^2}{r} \), where \(m\) is the mass of the proton (approximately \(1.67 \times 10^{-27} \text{ kg}\)).
04

Insert Values and Compute

Insert the known values into the centripetal force formula: \[ F = \frac{(1.67 \times 10^{-27} \text{ kg})(2.25 \times 10^8 \text{ m/s})^2}{628 \text{ m}} \] Simplifying gives: \[ F = \frac{1.67 \times 10^{-27} \times 5.0625 \times 10^{16}}{628} \approx 1.35 \times 10^{-10} \text{ N} \].
05

State the Result

The net force on the proton due to its circular motion is approximately \(1.35 \times 10^{-10} \text{ N}\).

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Key Concepts

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

Understanding a Particle Accelerator
A particle accelerator is a complex machine that propels charged particles, like protons, to high speeds. This is achieved using electric fields that increase the particles' velocities along a designed path. The main purpose of these accelerators is to investigate fundamental particles by creating high-energy collisions. The setup involves guiding particles into circular or linear paths where they achieve speeds close to the speed of light, denoted as "c".
  • They can be linear or circular.
  • Used in research, medicine, and industry.
  • Produce high-energy particles for experiments.
The path is often circular, especially in large machines, like the Large Hadron Collider. By making particles collide at high speeds, scientists can explore the components of the universe at a subatomic level.
Proton Speed in Particle Accelerators
Protons in accelerators are propelled to speeds approaching that of light. They need immense energy to reach such velocities due to their mass. The speed of protons within these machines is often a fraction of the speed of light. In the given exercise, the proton's speed is 0.750 times the speed of light.
This can be calculated as: \[ v = 0.750c = 0.750 \times 3.00 \times 10^8 \, \text{m/s} \]which gives a proton speed of \( 2.25 \times 10^8 \text{ m/s} \).
  • Speed is a direct product of accelerating mechanisms.
  • High speeds facilitate particle collisions.
  • Speeds expressed as a fraction of light speed to denote relativity effects.
Protons at this speed possess significant kinetic energy, necessary for particle collision experiments.
Circular Motion In a Particle Accelerator
Circular motion refers to a particle moving along a circular path at a constant speed. In accelerators, protons are maintained in such paths using electromagnetic fields. This ensures they travel consistently without veering off course.
The exercise describes a proton in an accelerator moving in a circular path with a radius of 628 meters. Its path stays circular due to centripetal force鈥攖he force pulling the proton towards the center of its path.
  • Circular paths provide stability for controlled acceleration.
  • Centroidal containment using strong magnetic fields.
  • Uniform speeds maintained by synchronizing energy input.
Circular motion is crucial to maintaining high velocities while minimizing energy loss due to deviation from the desired trajectory.
Net Force Calculation in Circular Motion
The net force experienced by a particle in circular motion is called centripetal force. It acts inward, pulling the particle towards the circle's center, thus maintaining the circular path.
To compute this force for our proton, use the formula:\[ F = \frac{mv^2}{r} \] where:
  • \( m \) is the particle's mass (\(1.67 \times 10^{-27} \text{ kg}\) for a proton).
  • \( v \) is the speed of the particle (\( 2.25 \times 10^8 \text{ m/s} \)).
  • \( r \) is the radius of the circle (628 m).
Substituting these values in gives:\[ F = \frac{(1.67 \times 10^{-27} \times 5.0625 \times 10^{16})}{628} \approx 1.35 \times 10^{-10} \text{ N} \, \]indicating the force necessary to change the proton's direction while maintaining its speed.

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Most popular questions from this chapter

A spaceship flies past Mars with a speed of 0.985\(c\) relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0\(\mu\) s. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Electrons are accelerated through a potential difference of \(750 \mathrm{kV},\) so that their kinetic energy is \(7.50 \times 10^{5} \mathrm{eV}\) . (a) What is the ratio of the speed \(v\) of an electron having this energy to the speed of light, \(c ?\) (b) What would the speed be if it were computed from the principles of classical mechanics?

Albert in Wonderland. Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 \(\mathrm{m}\) from each other. Being very skilled players, they play without a net. The tennis ball has mass 0.0580 \(\mathrm{kg} .\) You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at 80.0 \(\mathrm{m} / \mathrm{s} .\) What is the ball's kinetic energy? (b) Einstein slams a return at \(1.80 \times 10^{8} \mathrm{m} / \mathrm{s}\) . What is the ball's kinetic energy? (c) During Einstein's return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of \(2.20 \times 10^{8} \mathrm{m} / \mathrm{s}\) relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run \(20.0 \mathrm{m},\) according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

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

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression \(m v ?\) Express your answer in terms of the speed of light. (b) A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.
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