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Chapter 13: Problem 32

An object is released from rest at an altitude \(h\) above the surface of the Earth. (a) Show that its speed at a distance \(r\) from the Earth's center, where \(R_{E} \leq r \leq R_{E}+h,\) is given by $$v=\sqrt{2 G M_{E}\left(\frac{1}{r}-\frac{1}{R_{E}+h}\right)}$$ (b) Assume the release altitude is \(500 \mathrm{km} .\) Perform the integral $$\Delta t=\int_{i}^{f} d t=-\int_{i}^{f} \frac{d r}{v}$$ to find the time of fall as the object moves from the release point to the Earth's surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is \(v=-d r / d t\). Perform the integral numerically.

Short Answer

Expert verified
Use of energy conservation principle to prove the velocity equation was successful. The time integral is then evaluated via numerical methods by substituting the velocity formula. This integral provides the time taken by the object to fall to the earth's surface from the release point.

Step by step solution

01

Analyze given speed equation

The given equation is essentially expressing the energy conservation law for the object in motion. Gravitational potential energy is converted into kinetic energy as the object falls.
02

Prove the speed equation

We start with the energy conservation equation: \(mgh = \frac{1}{2} mv^{2} + (-\frac{G M_{E} m}{r})\). Solving with given condition we get: \( \frac{1}{2} mv^{2} = G M_{E} m (\frac{1}{r}-\frac{1}{R_{E}+h})\). Further simplification results into: \(v= \sqrt{2 G M_{E}\left(\frac{1}{r}-\frac{1}{R_{E}+h}\right)}.\) The last equation is the formula for the velocity that we needed to prove.
03

Analysis of the fall time integral

The given integral is used to compute the time the object takes to fall from the release point to the surface of the Earth. We will essentially be solving the integral, \(\Delta t=\int_{i}^{f} \frac{d r}{-v}\). To carry out this integral, we need to replace \(v\) with its formula from the first part of the problem. This will result in an integral that might be difficult to solve analytically, hence numerical methods of integration may be necessary.
04

Carry out the integral numerically

Substitute the speed formula into the integral, and solve it numerically using computational software. Given the complexity of the integral, buffer and numerical libraries like SciPy or NumPy in python could be used to solve it. Keep in mind that the initial boundary \(i\) is \(R_{E} + h\) and the final boundary \(f\) is \(R_{E}\). This is because the object is moving from a height h above the Earth's surface, towards the Earth's surface.

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Key Concepts

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

Energy Conservation
Energy conservation is one of the fundamental principles in physics. It tells us that energy cannot be created or destroyed, only converted from one form to another. When dealing with gravitational physics, this principle becomes especially significant.

In the context of the exercise, as an object falls from a height down to the Earth, its gravitational potential energy is converted into kinetic energy. The potential energy at height is given by the formula:
  • Gravitational Potential Energy (GPE): \( mgh \)
However, because the object is at a distance from the center of the Earth, a more accurate expression using gravitational parameters would be:
  • \( -\frac{G M_E m}{r} \)
where:- \( G \) is the gravitational constant.- \( M_E \) is Earth's mass.- \( m \) is the mass of the object.- \( r \) is the distance from the center of the Earth.
As the object falls, this energy changes form to kinetic energy (the energy of motion) when it hits the Earth's surface:
  • Kinetic Energy (KE): \( \frac{1}{2} mv^2 \)
You can see in the exercise that through manipulation of these energy expressions, we derive the required speed equation.
Kinetic Energy
Kinetic energy refers to the energy possessed by a moving object. The faster an object moves, the more kinetic energy it has. This energy is calculated using the formula:
  • \( KE = \frac{1}{2} mv^2 \)
where:- \( m \) is the mass of the object.- \( v \) is the velocity of the object.
In our exercise, we see that as the object falls towards the Earth, the speed of the object increases, meaning more kinetic energy is being accumulated. This is in line with the energy conservation principle where gravitational potential energy converts into kinetic energy.

The object started from rest, as mentioned in the problem, so its initial kinetic energy was zero. After being released, the potential energy is gradually converted into kinetic energy, hence increasing the speed as it moves closer to Earth's surface. This process is seamless and continuous without loss of energy, thanks to the conservation law. The expression for speed derived in the exercise perfectly encapsulates this conversion by linking gravitational parameters to kinetic energy.
Numerical Integration
When solving integrals analytically is too complex, numerical integration offers a practical approach. It involves estimating the value of an integral using approximation methods rather than finding an exact formula.

In the exercise, the given integral \( \Delta t = \int_{i}^{f} \frac{dr}{-v} \) is used to calculate the time it takes for the object to fall from its initial height to Earth's surface. This involves the complex speed function derived earlier. Due to its complexity, using computational tools like Python libraries SciPy or NumPy is recommended for solving it numerically.

When using numerical integration:
  • Ensure that the boundaries of integration are correct; here, they move from \( R_E + h \) to \( R_E \).
  • Choose an appropriate numerical method, like the trapezoidal rule or Simpson鈥檚 rule, which these libraries facilitate.
  • Remember that the accuracy of a numerical solution depends on the method and step size used.
This ensures that we estimate the falling time accurately, bridging the gap between complicated integrals and practical solutions.

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Most popular questions from this chapter

(a) \(A 5.00-k g\) object is released \(1.20 \times 10^{7} \mathrm{m}\) from the center of the Earth. It moves with what acceleration relative to the Earth? (b) What If? A \(2.00 \times 10^{24} \mathrm{kg}\) object is released \(1.20 \times 10^{7} \mathrm{m}\) from the center of the Earth. It moves with what acceleration relative to the Earth? Assume that the objects behave as pairs of particles, isolated from the rest of the Universe.

Newton's law of universal gravitation is valid for distances covering an enormous range, but it is thought to fail for very small distances, where the structure of space itself is uncertain. Far smaller than an atomic nucleus, this crossover distance is called the Planck length. It is determined by a combination of the constants \(G, c,\) and \(h\) where \(c\) is the speed of light in vacuum and \(h\) is Planck's constant (introduced in Chapter 11 ) with units of angular momentum. (a) Use dimensional analysis to find a combination of these three universal constants that has units of length. (b) Determine the order of magnitude of the Planck length. You will need to consider noninteger powers of the constants.

The acceleration of an object moving in the gravitational field of the Earth is $$\mathbf{a}=-\frac{G M_{E} \mathbf{r}}{r^{3}}$$ where \(r\) is the position vector directed from the center of the Farth toward the object. Choosing the origin at the center of the Earth and assuming that the small object is moving in the \(x y\) plane, we find that the rectangular (Cartesian) components of its acceleration are $$a_{x}=-\frac{G M_{E} x}{\left(x^{2}+y^{2}\right)^{3 / 2}} \quad a_{y}=-\frac{G M_{E} y}{\left(x^{2}+y^{2}\right)^{3 / 2}}$$ Use a computer to set up and carry out a numerical prediction of the motion of the object, according to Euler's method. Assume the initial position of the object is \(x=0\) and \(y=2 R_{E},\) where \(R_{E}\) is the radius of the Earth. Give the object an initial velocity of \(5000 \mathrm{m} / \mathrm{s}\) in the \(x\) direction. The time increment should be made as small as practical. Try 5 s. Plot the \(x\) and \(y\) coordinates of the object as time goes on. Does the object hit the Earth? Vary the initial velocity until you find a circular orbit.

A certain quaternary star system consists of three stars, each of mass \(m,\) moving in the same circular orbit of radius \(r\) about a central star of mass \(M .\) The stars orbit in the same sense, and are positioned one third of a revolution apart from each other. Show that the period of each of the three stars is given by $$T=2 \pi \sqrt{\frac{r^{3}}{g(M+m / \sqrt{3})}}$$

In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant \(G\) uses lead spheres with masses of \(1.50 \mathrm{kg}\) and \(15.0 \mathrm{g}\) whose centers are separated by about \(4.50 \mathrm{cm} .\) Calculate the gravitational force between these spheres, treating each as a particle located at the center of the sphere.
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