/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 One of the most studied objects ... [FREE SOLUTION] | 91Ó°ÊÓ

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One of the most studied objects in the night sky is the Crab nebula, the remains of a supernova explosion observed by the Chinese in \(1054 . \ln 1968\) it was discovered that a pulsar-a rapidly rotating neutron star that emits a pulse of radio waves with each revolution-lies near the center of the Crab nebula. The period of this pulsar is 33 ms. What is the angular speed (in rad/s) of the Crab nebula pulsar?

Short Answer

Expert verified
The angular speed of the Crab nebula pulsar is approximately 190.42 rad/s.

Step by step solution

01

Understand the Problem

We are given the period of the pulsar as 33 milliseconds and need to find its angular speed in radians per second.
02

Identify the Period in Seconds

Convert the period from milliseconds to seconds. Since 1 second equals 1000 milliseconds, the conversion is as follows:\[T = 33 \text{ ms} = 33 \times 10^{-3} \text{ s} = 0.033 \text{ s}\]
03

Use the Formula for Angular Speed

Angular speed is calculated using the formula:\[\omega = \frac{2\pi}{T}\]where \(\omega\) is the angular speed and \(T\) is the period in seconds.
04

Plug in the Values to Calculate Angular Speed

Substitute the period into the angular speed formula:\[\omega = \frac{2\pi}{0.033}\]Calculate the value:\[\omega \approx 190.42 \text{ rad/s}\]
05

Finalize the Answer

The angular speed of the Crab nebula pulsar is approximately 190.42 radians per second.

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Key Concepts

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

Pulsar
A pulsar is an incredible astronomical phenomenon that captures the attention of scientists and space enthusiasts alike. Pulsars are highly magnetized, spinning neutron stars that emit beams of electromagnetic radiation from their magnetic poles. As these stars rotate, the radiation beams are observed from Earth as pulses of radio waves, hence the name "pulsar."

A pulsar acts like a cosmic lighthouse with its beams observable each time they sweep across our line of sight. These rotations can occur extremely rapidly, with some pulsars completing a full spin multiple times each second. The precision of their pulse intervals makes pulsars excellent cosmic clocks. However, understanding pulsars involves delving into their origin, which ties into the broader lifecycle of stars. To truly grasp what a pulsar is, it's important to understand what neutron stars are and how they form.
Neutron Star
Neutron stars are the remnants of massive stars that have ended their life cycle in a catastrophic supernova explosion. These stars are incredibly dense core remnants, formed when the core collapses under its own gravity after all nuclear fuel is exhausted.

The gravity of the collapsing core is so immense that protons and electrons combine to form neutrons, resulting in a star composed almost entirely of neutrons, hence the name "neutron star." These celestial objects are just about 20 kilometers in diameter yet pack a mass exceeding that of the Sun. Their density is astonishing; a sugar-cube-sized amount of neutron-star material would have a mass equal to that of all humanity.

When conditions are right, neutron stars can become pulsars. Their super-rapid rotation and strong magnetic fields produce pulses of radiation that make pulsars easy to observe from Earth.
Supernova Explosion
A supernova explosion is one of the universe's most spectacular and energetic events. It occurs during the last stages of a massive star's life, as the star runs out of nuclear fuel and the balance between pressure and gravity shifts dramatically.

Once the star's core collapses, it triggers an explosion so violent that it can outshine entire galaxies for a short time. The explosion disperses the elements the star has fused over time into space, seeding the cosmos with the ingredients for new star systems and planets.
  • The exploding star can shine billions of times brighter than the Sun.
  • Observed remnants from these explosions can provide crucial information about the composition of the universe.
  • Supernovas play a key role in the formation of neutron stars or black holes.
The study of supernova explosions grants astronomers insights into the origins of neutron stars and features like the Crab nebula.
Crab Nebula
The Crab nebula is an iconic astronomical object formed from the remnants of a supernova explosion recorded in 1054. Located in the Taurus constellation, it's one of the most studied and well-documented nebulae in the night sky.

The Crab nebula serves as a beautiful and intricate cosmic tapestry woven from gasses and particles ejected by the supernova. Its core harbors a pulsar, providing a live demonstration of the key outcome of stellar evolution. This pulsar's rapid rotation and radio wave emission offer astronomers a chance to study the physics of neutron stars and magnetic fields.
  • The Crab nebula is visible through moderate-sized telescopes as an expanding cloud of light.
  • It is a source of cosmic rays and electromagnetic radiation across the spectrum.
  • Its pulsar is noted for its precision, contributing to our understanding of pulsars as cosmic clocks.
These studies make the Crab nebula a vital source of information about the lifecycle of stars, particularly in understanding the transition from supernova remnants to neutron stars.

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Most popular questions from this chapter

A ceiling fan is rotating at 0.96 rev /s. When turned off, it slows uniformly to a stop in \(2.4 \mathrm{min}\). (a) How many revolutions does the fan make in this time? (b) Using the result from part (a), find the number of revolutions the fan must make for its speed to decrease from 0.96 rev \(/ \mathrm{s}\) to \(0.48 \mathrm{rev} / \mathrm{s}\) .

When a compact disk with a 12.0 -cm diameter is rotating at 5.05 rad \(/ \mathrm{s},\) what are (a) the linear speed and (b) the centripetal acceleration of a point on its outer rim? (c) Consider a point on the CD that is halfway between its center and its outer rim. Without repeating all of the calculations required for parts (a) and (b), determine the linear speed and the centripetal acceleration of this point.

A basketball rolls along the floor with a constant linear speed \(v .\) (a) Find the fraction of its total kinetic energy that is in the form of rotational kinetic energy about the center of the ball. (b) If the linear speed of the ball is doubled to \(2 v,\) does your answer to part (a) increase, decrease, or stay the same? Explain.

A compact disk (CD) speeds up uniformly from rest to 310 rpm in \(3.3 \mathrm{s}\). (a) Describe a strategy that allows you to calculate the number of revolutions the CD makes in this time. (b) Use your strategy to find the number of revolutions.

The drill used by most dentists today is powered by a small air turbine that can operate at angular speeds of 350,000 rpm. These drills, along with ultrasonic dental drills, are the fastest turbines in the world-far exceeding the angular speeds of jet engines. Suppose a drill starts at rest and comes up to operating speed in \(2.1 \mathrm{s}\). (a) Find the angular acceleration produced by the drill, assuming it to be constant. (b) How many revolutions does the drill bit make as it comes up to speed?

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