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Chapter 13: Problem 21

Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev/s. If such a star has a radius of \(20 \mathrm{~km}\), what must be its minimum mass so that material on its surface remains in place during the rapid rotation?

Short Answer

Expert verified
The minimum mass of the neutron star is approximately \(4.734 \times 10^{28}\) kg.

Step by step solution

01

Understand the Problem

We need to find the minimum mass of the neutron star so that the gravitational force is sufficient to counteract the centrifugal force acting on a piece of material at its surface during its rotation.
02

Identify Relevant Formulas

We will use the formula for gravitational force, \( F_g = \frac{G M m}{R^2} \), and for centrifugal force, \( F_c = m \omega^2 R \). For equilibrium, \( F_g = F_c \). Here, \( G \) is the gravitational constant, \( M \) is the mass of the star, \( m \) is the mass of the material, \( R \) is the radius, and \( \omega \) is the angular velocity.
03

Determine Angular Velocity

The star rotates at 1 revolution per second. We need to convert this to angular velocity in radians per second: \( \omega = 2\pi \times 1 = 2\pi \text{ rad/s} \).
04

Set Up the Equation

Set the gravitational force equal to the centrifugal force: \[ \frac{G M m}{R^2} = m \omega^2 R \]. Simplifying gives \[ G M = \omega^2 R^3 \]. Substitute \( G = 6.67 \times 10^{-11} \text{ N m}^2 \text{/kg}^2 \), \( \omega = 2\pi \text{ rad/s} \), and \( R = 20,000 \text{ m} \) (since 20 km = 20,000 m).
05

Solve for Mass \( M \)

Plug the values into the equation \[ G M = \omega^2 R^3 \]: \[ 6.67 \times 10^{-11} M = (2\pi)^2 (20,000)^3 \]. Solve for \( M \): \[ M = \frac{(2\pi)^2 (20,000)^3}{6.67 \times 10^{-11}} \].
06

Calculate the Mass

Perform the calculation: \( M \approx \frac{39.4784 \times 8 \times 10^{12}}{6.67 \times 10^{-11}} = \frac{3.158272 \times 10^{14}}{6.67 \times 10^{-11}} \approx 4.734 \times 10^{28} \text{ kg} \).

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Key Concepts

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

Gravitational Force
Gravitational force is a fundamental force that attracts two masses towards each other. In the context of a neutron star, it is the force that keeps material on the star's surface from flying off into space.
The formula to calculate gravitational force is given by:
  • \( F_g = \frac{G M m}{R^2} \)
- \( G \) is the gravitational constant, approximately \( 6.67 \times 10^{-11} \text{ N m}^2 \text{/kg}^2 \).
- \( M \) is the mass of the neutron star.- \( m \) is the mass of the material on the surface.- \( R \) is the radius of the neutron star.This force is essential to counteracting the centrifugal force that results from the star's rotation. Without sufficient gravitational force, the material would not stay attached to the rapidly spinning surface.
Centrifugal Force
Centrifugal force is the apparent force that seems to push a rotating object away from the center of rotation. In our neutron star scenario, this force acts outward on any material on the star's surface.
Centrifugal force is given by the formula:
  • \( F_c = m \omega^2 R \)
- \( m \) is the mass of the surface material.- \( \omega \) is the angular velocity (rate of rotation) of the star.- \( R \) is the radius of the star.The basic principle here is that as the neutron star spins, every bit of mass on its surface experiences this outward force. For the material to remain on the surface, the gravitational force must be equal to or greater than the centrifugal force.
Angular Velocity
Angular velocity is a measure of how fast something rotates or spins. It tells us how many radians an object rotates in a certain amount of time.
For the neutron star, the angular velocity \( \omega \) is calculated based on its rotational speed. Given that the star makes one full revolution per second, its angular velocity is:
  • \( \omega = 2\pi \text{ rad/s} \)
This conversion comes from the fact that one complete revolution equals \( 2\pi \) radians. Angular velocity plays a key role in calculating centrifugal force and is crucial to determining whether the gravitational pull of the neutron star is enough to prevent the material from being flung away by the centrifugal force.

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Most popular questions from this chapter

An object of mass \(m\) is initially held in place at radial distance \(r=3 R_{E}\) from the center of Earth, where \(R_{E}\) is the radius of Earth. Let \(M_{E}\) be the mass of Earth. A force is applied to the object to move it to a radial distance \(r=4 R_{E}\), where it again is held in place. Calculate the work done by the applied force during the move by integrating the force magnitude.

At what altitude above Earth's surface would the gravitational acceleration be \(4.9 \mathrm{~m} / \mathrm{s}^{2} ?\)

In a shuttle craft of mass \(m=3000 \mathrm{~kg}\), Captain Janeway orbits a planet of mass \(M=9.50 \times 10^{25} \mathrm{~kg}\), in a circular orbit of radius \(r=4.20 \times 10^{7} \mathrm{~m} .\) What are (a) the period of the orbit and (b) the speed of the shuttle craft? Janeway briefly fires a forwardpointing thruster, reducing her speed by \(2.00 \%\). Just then, what are (c) the speed, (d) the kinetic energy, (e) the gravitational potential energy, and (f) the mechanical energy of the shuttle craft? (g) What is the semimajor axis of the elliptical orbit now taken by the craft? (h) What is the difference between the period of the original circular orbit and that of the new elliptical orbit? (i) Which orbit has the smaller period?

In his 1865 science fiction novel From the Earth to the Moon, Jules Verne described how three astronauts are shot to the Moon by means of a huge gun. According to Verne, the aluminum capsule containing the astronauts is accelerated by ignition of nitrocellulose to a speed of \(11 \mathrm{~km} / \mathrm{s}\) along the gun barrel's length of \(220 \mathrm{~m} .\) (a) In \(g\) units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts? A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston along the gun's tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves \(3.5 \mathrm{~km}\) and reaches a speed of \(7.0 \mathrm{~km} / \mathrm{s}\). Once launched, the rocket can be fired to gain additional speed. (c) In \(g\) units, what would be the average acceleration of the rocket within the launcher? (d) How much additional speed is needed (via the rocket engine) if the rocket is to orbit Earth at an altitude of \(700 \mathrm{~km} ?\)

(a) At what height above Earth's surface is the energy required to lift a satellite to that height equal to the kinetic energy required for the satellite to be in orbit at that height? (b) For greater heights, which is greater, the energy for lifting or the kinetic energy for orbiting?
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