/*! 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 32 A neutron star has a mass of \(2... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 32

A neutron star has a mass of \(2.0 \times 10^{30} \mathrm{~kg}\) (about the mass of our sun) and a radius of \(5.0 \times 10^{3} \mathrm{~m}\) (about the height of a goodsized mountain). Suppose an object falls from rest near the surface of such a star. How fast would it be moving after it had fallen a distance of \(0.010 \mathrm{~m} ?\) (Assume that the gravitational force is constant over the distance of the fall, and that the star is not rotating.)

Short Answer

Expert verified
The object would be moving at approximately \( 3.267 \times 10^5 \ \mathrm{m/s} \) after falling 0.010 m.

Step by step solution

01

Identify the Known Values

We need to calculate the velocity of an object after it falls a distance of 0.010 m on a neutron star. The mass of the neutron star is given as \( 2.0 \times 10^{30} \) kg and the radius is \( 5.0 \times 10^{3} \) m. The object starts from rest.
02

Calculate Gravitational Acceleration

Use the formula for gravitational force to find the acceleration due to gravity on the neutron star's surface: \[ g = \frac{G M}{R^2} \]where \( G = 6.674 \times 10^{-11} \ \mathrm{Nm}^2/\mathrm{kg}^2 \) is the gravitational constant, \( M = 2.0 \times 10^{30} \) kg is the mass of the neutron star, and \( R = 5.0 \times 10^{3} \) m is its radius. Substituting these values in gives:\[ g = \frac{(6.674 \times 10^{-11}) \times (2.0 \times 10^{30})}{(5.0 \times 10^{3})^2} \approx 5.3392 \times 10^{12} \ \mathrm{m/s^2} \]
03

Calculate Final Velocity

We use the kinematic equation connecting initial velocity, acceleration, and distance to find the final velocity:\[ v^2 = u^2 + 2as \]where \( u = 0 \ \mathrm{m/s} \) is the initial velocity, \( a \) is the acceleration due to gravity calculated in Step 2, and \( s = 0.010 \) m is the distance fallen. Substituting these values gives:\[ v^2 = 0 + 2 \cdot 5.3392 \times 10^{12} \cdot 0.010 \]\[ v^2 = 1.06784 \times 10^{11} \]Taking the square root gives the final velocity:\[ v \approx 3.267 \times 10^5 \ \mathrm{m/s} \]
04

Concluding Result

The final velocity of the object after falling 0.010 m on the neutron star's surface is approximately \( 3.267 \times 10^5 \ \mathrm{m/s} \). This high velocity is due to the strong gravitational field of the neutron star.

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Key Concepts

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

Kinematics
Kinematics is the field of physics that describes the motion of objects without considering the forces that cause the motion. It focuses on the movement regarding parameters like displacement, velocity, and acceleration. In the exercise presented, the kinematic equations have been employed to determine how an object accelerates and reaches a particular velocity as it falls. These equations are crucial in describing the behavior of any object in motion, providing a mathematical framework that helps predict and analyze the object's path.

Kinematics involves several key concepts:
  • Displacement: A vector quantity that refers to the change in position of an object.
  • Velocity: The rate at which an object changes its position.
  • Acceleration: The rate of change of velocity over time.
A fundamental understanding of these concepts is essential in solving any kinematic problem 鈥� whether on Earth or in more extreme environments like a neutron star.
Gravitational Acceleration
Gravitational acceleration refers to the acceleration of an object due to the influence of a gravitational force. On Earth, we experience a gravitational acceleration of approximately 9.8 m/s虏. However, when we examine celestial bodies such as a neutron star, this acceleration is drastically different due to its immense mass and small radius.

In our exercise, the gravitational acceleration on the surface of the neutron star is calculated using the formula:\[ g = \frac{G M}{R^2} \]where:
  • G is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \mathrm{Nm}^2/\mathrm{kg}^2 \).
  • M is the mass of the neutron star.
  • R is the radius of the neutron star.
This results in an acceleration of approximately \( 5.3392 \times 10^{12} \, \mathrm{m/s^2} \), which is vastly higher than what is experienced on Earth and illustrates the powerful gravitational pull exerted by a neutron star.
Neutron Star
A neutron star is a highly dense celestial object that remains after a massive star explodes in a supernova. These stars have incredibly high densities because they are composed almost entirely of neutrons packed closely together.

Some fascinating characteristics of neutron stars include:
  • They can have masses greater than the Sun while being only about 20 kilometers in diameter.
  • Their gravitational fields are immensely powerful due to the combination of high mass and small size.
  • Neutron stars can rotate at extremely high speeds.
In our problem, we're dealing with the gravitational effects of a neutron star, which are significant due to its massive gravitational forces. This exercise provides insights into how such a strong gravitational field affects an object's movement, even when falling a short distance.
Kinematic Equations
Kinematic equations are a set of mathematics formulas used to predict the future position or speed of an object moving under constant acceleration. These equations help to extract quantitative information about motion along a straight line.

The fundamental kinematic equation used in our exercise is:\[ v^2 = u^2 + 2as \]where:
  • v is the final velocity of the object.
  • u is the initial velocity.
  • a is the acceleration acting on the object.
  • s is the displacement.
This equation enabled us to compute the object's speed after falling 0.010 m on the neutron star. Understanding how to manipulate and apply these kinematic equations is essential in solving physics problems, especially when analyzing motion in extraordinary environments such as the intense gravitational field of a neutron star.

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Most popular questions from this chapter

A toboggan slides down a hill and has a constant velocity. The angle of the hill is \(8.00^{\circ}\) with respect to the horizontal. What is the coefficient of kinetic friction between the surface of the hill and the toboggan?

An arrow, starting from rest, leaves the bow with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). If the average force exerted on the arrow by the bow were doubled, all else remaining the same, with what speed would the arrow leave the bow?

A person is attempting to push a refrigerator across a room. He exerts a horizontal force on the refrigerator, but it does not move. (a) What other horizontal force must be acting on the refrigerator? (b) How are the magnitude and direction of this force related to the force that the person exerts? (c) Suppose that the person applies the force such that it is the largest possible force before the refrigerator begins to move. What factors determine the magnitude of this force? Problem Consult Multiple-Concept Example 9 to explore a model for solving this problem. A person pushes on a 57 -kg refrigerator with a horizontal force of \(-267 \mathrm{~N}\); the minus sign indicates that the force is directed along the \(-x\) direction. The coefficient of static friction is \(0.65 .\) (a) If the refrigerator does not move, what is the magnitude and direction of the static frictional force that the floor exerts on the refrigerator? (b) What is the magnitude of the largest pushing force that can be applied to the refrigerator before it just begins to move?

The space probe Deep Space 1 was launched on October 24,1998 . Its mass was \(474 \mathrm{~kg}\). The goal of the mission was to test a new kind of engine called an ion propulsion drive. This engine generated only a weak thrust, but it could do so over long periods of time with the consumption of only small amounts of fuel. The mission was spectacularly successful. At a thrust of \(56 \mathrm{mN}\) how many days were required for the probe to attain a velocity of \(805 \mathrm{~m} / \mathrm{s}(1800 \mathrm{mi} / \mathrm{h})\), assuming that the probe started from rest and that the mass remained nearly constant?

On earth, two parts of a space probe weigh \(11000 \mathrm{~N}\) and \(3400 \mathrm{~N}\). These parts are separated by a center-to-center distance of \(12 \mathrm{~m}\) and may be treated as uniform spherical objects. Find the magnitude of the gravitational force that each part exerts on the other out in space, far from any other objects.
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