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Chapter 5: Problem 14

Gravitational pull of a uniform sheet A uniform square sheet of metal is floating motionless in space: The sheet is \(10 \mathrm{~m}\) on a side and of negligible thickness, and it has a mass of 10 metric tonnes. a) Consider the gravitational force due to the plate felt by a point mass of \(1 \mathrm{~kg}\) a distance \(z\) from the center of the square, in the direction perpendicular to the sheet, as shown above. Show that the component of the force along the \(z\)-axis is $$ F_{z}=G \sigma z \iint_{-L / 2}^{L / 2} \frac{\mathrm{d} x \mathrm{~d} y}{\left(x^{2}+y^{2}+z_{2}^{2}\right)^{3 / 2}}, $$ where \(G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant and \(\sigma\) is the mass per unit area of the sheet. b) Write a program to calculate and plot the force as a function of \(z\) from \(z=0\) to \(z=10 \mathrm{~m}\). For the double integral use (double) Gaussian quadrature, as in Eq. (5.82), with 100 sample points along each axis. c) You should see a smooth curve, except at very small values of \(z\), where the force should drop off suddenly to zero. This drop is not a real effect, but an artifact of the way we have done the calculation. Explain briefly where this artifact comes from and suggest a strategy to remove it, or at least to decrease its size. This calculation can thought of as a model for the gravitational pull of a galaxy. Most of the mass in a spiral galaxy (such as our own Milky Way) lies in a thin plane or disk passing through the galactic center, and the gravitational pull exerted by that plane on bodies outside the galaxy can be calculated by just the methods we have employed here.

Short Answer

Expert verified
The integral for the gravitational force is simplified using mass density and Gaussian quadrature. For small z, the numerical method's inaccuracies cause a drop in the force, which can be mitigated by increasing sample points or using adaptive quadrature.

Step by step solution

01

- Understanding the parameters

The problem involves calculating the gravitational force exerted by a uniform square sheet of metal of size 10 m on a side, and mass 10 metric tonnes (10,000 kg), on a point mass of 1 kg located at a distance z along the perpendicular axis to the sheet.
02

- Expression for mass per unit area (σ)

Given the mass of the sheet is 10,000 kg and the area of the sheet is \(A = L^2 \), where \(L = 10 \, m \). Therefore, the mass per unit area is \(\sigma = \frac{M}{A} = \frac{10,000}{10^2} = 100 \, kg/m^2 \).
03

- Simplifying the integral for gravitational force

The gravitational force along the z-axis can be expressed using Newton's law of gravitation. Given the density of the plate is uniform, this allows us to rewrite the integral:\[F_z = G \sigma z \iint_{-L/2}^{L/2} \frac{ \, dx \, dy}{\left(x^2+y^2+z^2\right)^{3/2}}\]where \( G = 6.674 \times 10^{-11} \, N\cdot m^2 / kg^2 \).
04

- Implementing Gaussian quadrature

Write a program using Python or any suitable language to evaluate the double integral. Use Gaussian quadrature with 100 sample points along each axis to calculate the integral. This method will approximate the integral by taking weighted sums of function values at specific points.
05

- Plotting the force as a function of z

Calculate the force for varying values of z from 0 to 10 meters. Use a plotting library such as Matplotlib in Python to plot the value of the force %%% versus the distance z.
06

- Analysis of results for small z values

At very small values of z, you might notice a sudden drop in the force. This artifact occurs because our numerical integration method has lower accuracy near singularities (when z is very close to 0). To mitigate this, one can increase the number of sample points or use an adaptive quadrature method which applies more sample points near the singularity.

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Key Concepts

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

Gaussian Quadrature
Gaussian quadrature is a numerical integration method used to approximate the value of definite integrals, especially when traditional analytical solutions are difficult or impossible to find. It works by evaluating the integrand at specific points called *nodes*, and weighing these values appropriately. The method can achieve high accuracy with relatively few function evaluations.

In the context of calculating the gravitational force exerted by the metal sheet, Gaussian quadrature helps us evaluate the double integral: \[ F_z = G \sigma z \iint_{-L/2}^{L/2} \frac{d x \ d y}{(x^2 + y^2 + z^2)^{3/2}} \].

To use Gaussian quadrature effectively:
  • Choose the number of sample points along each axis. In this problem, 100 points were used.
  • Calculate the nodes (where the function is evaluated) and weights (how much each function value counts toward the sum).
  • Sum the weighted function values to approximate the double integral.
This approach ensures a smooth and accurate approximation for the integral over the square sheet.
Newton's Law of Gravitation
Newton's law of gravitation states that every point mass in the universe attracts every other point mass with a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is:
\[ F = G \frac{m1 \times m2}{r^2} \].

In the exercise given, this principle is adapted to calculate the gravitational force exerted by a continuous mass distribution (the sheet) on a point mass.

Here, the force in the z-direction due to the entire sheet is found by integrating the contributions of every infinitesimal mass element of the sheet. The key steps involve:
  • Calculating the mass per unit area (\sigma) of the sheet,
  • Expressing the gravitational force differential and integrating it over the entire sheet.
This integration accounts for the fact that each small element of mass exerts a force according to Newton's law, and combining these gives the total gravitational force.
Numerical Integration
Numerical integration is a broad category of techniques used to estimate the value of integrals when an analytical solution is difficult or impossible to obtain. Popular methods include the trapezoidal rule, Simpson's rule, and Gaussian quadrature.

This problem employs numerical integration to handle a double integral over a square area, to find the force exerted by a square sheet. Steps involved are:
  • Expressing the integral representing the gravitational force: \[ F_z = G \sigma z \iint_{-L/2}^{L/2} \frac{d x \ d y}{(x^2 + y^2 + z^2)^{3/2}} \],
  • Selecting a numerical method like Gaussian quadrature to approximate the integral.
The accuracy of numerical integration can be influenced by:
  • The number of sample points (more points generally result in higher accuracy).
  • The handling of singularities (values where the integrand becomes infinite or undefined).
Addressing the artifact at low z values involves the concept of adaptive quadrature, which increases sample points in regions where the function changes rapidly, such as near singularities.

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Most popular questions from this chapter

Electric tield of a charge distribution: Suppose we have a distribution of charges and we want to calculate the resulting electric field. One way to do this is to first calculate the electric potential \(\phi\) and then take its gradient. For a point charge \(q\) at the origin, the electric potential at a distance \(r\) from the origin is \(\phi=q / 4 \pi \epsilon_{0} r\) and the electric field is \(\mathbf{E}=-\nabla \phi\). a) You have two charges, of \(\pm 1 \mathrm{C}, 10 \mathrm{~cm}\) apart. Calculate the resulting electric potential on a \(1 \mathrm{~m} \times 1 \mathrm{~m}\) square plane surrounding the charges and passing through them. Calculate the potential at \(1 \mathrm{~cm}\) spaced points in a grid and make a visualization on the screen of the potential using a density plot. b) Now calculate the partial derivatives of the potential with respect to \(x\) and \(y\) and hence find the electric field in the \(x y\) plane. Make a visualization of the field also. This is a little trickier than visualizing the potential, because the electric field has both magnitude and direction. One way to do it might be to make two density plots, one for the magnitude, and one for the direction, the latter using the "hsv"

Quantum uncertainty in the harmonic oscillator In units where all the constants are 1 , the wavefunction of the \(n\)th energy level of the one-dimensional quantum harmonic oscillator-i.e., a spinless point particle in a quadratic potential well-is given by $$ \psi_{n}(x)=\frac{1}{\sqrt{2^{11} n ! \sqrt{\pi}}} \mathrm{e}^{-x^{2} / 2} H_{n}(x), $$ for \(n=0 \ldots \infty\), where \(H_{n}(x)\) is the \(n\)th Hermite polynomial. Hermite polynomials satisfy a relation somewhat similar to that for the Fibonacci numbers, although more complex: $$ H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) . $$ The first two Hermite polynomials are \(H_{0}(x)=1\) and \(H_{1}(x)=2 x\). a) Write a user-defined function \(\mathrm{H}(\mathrm{n}, \mathrm{x})\) that calculates \(H_{n}(x)\) for given \(x\) and any integer \(n \geq 0\). Use your function to make a plot that shows the harmonic oscillator wavefunctions for \(n=0,1,2\), and 3, all on the same graph, in the range \(x=-4\) to \(x=4\). Hint: There is a function factorial in the ath package that calculates the factorial of an integer. b) Make a separate plot of the wavefunction for \(n=30\) from \(x=-10\) to \(x=10\). Hint If your program takes too long to run in this case, then you're doing the calculation wrong-the program should take only a second or so to run. c) The quantum uncertainty in the position of a particle in the \(n\)th level of a harmonic oscillator can be quantified by its root-mean-square position \(\sqrt{\left\langle x^{2}\right\rangle}\), where $$ \left\langle x^{2}\right\rangle=\int_{-\infty}^{\infty} x^{2}\left|\psi_{n}(x)\right|^{2} \mathrm{~d} x . $$ Write a program that evaluates this integral using Gaussian quadrature on 100 points, then calculates the uncertainty (i.e., the root-mean-square position of the particle) for a given value of \(n\). Use your program to calculate the uncertainty for \(n=5\). You should get an answer in the vicinity of \(\sqrt{\left\langle x^{2}\right\rangle}=2.3 .\)

The diffraction limit of a telescope Our ability to resolve detail in astronomical observation is limited by the diffraction of light in our telescopes. Light from stars can be treated effectively as coming from a point source at infinity. When such light, with wavelength \(\lambda\), passes through the circular aperture of a telescope (which we'll assume to have unit radius) and is focused by the telescope in the focal plane, it produces not a single dot, but a circular diffraction pattern consisting of central spot surrounded by a series of concentric rings. The intensity of the light in this diffraction pattern is given by $$ I(r)=\left(\frac{J_{1}(k r)}{k r}\right)^{2} $$ where \(r\) is the distance in the focal plane from the center of the diffraction pattern, \(k=2 \pi / \lambda\), and \(J_{1}(x)\) is a Bessel function. The Bessel functions \(J_{m}(x)\) are given by $$ J_{m}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (m \theta-x \sin \theta) \mathrm{d} \theta, $$ where \(m\) is a nonnegative integer and \(x \geq 0\). a) Write a Python function \(J(m, x)\) that calculates the value of \(J_{m}(x)\) using Simpson's rule with \(N=1000\) points. Use your function in a program to make a plot, on a single graph, of the Bessel functions \(J_{0}, J_{1}\) and \(J_{2}\) as a function of \(x\) from \(x=0\) to \(x=20\). b) Make a second program that makes a density plot of the intensity of the circular diffraction pattern of a point light source with \(\lambda=500 \mathrm{~nm}\), in a square region of the focal plane, using the formula given above. Your picture should cover values of \(r\) from zero up to about \(1 \mu \mathrm{m}\).

The Stefan-Boltzmann constant The Planck theory of thermal radiation tells us that in the (angular) frequency interval \(\omega\) to \(\omega+d \omega\), a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to \(I(\omega)\) d \(\omega\), where $$ I(\omega)=\frac{\hbar}{4 \pi^{2} c^{2}} \frac{\omega^{3}}{\left(\mathrm{e}^{\hbar \omega / k_{a} T}-1\right)} $$ Here \(h\) is Planck's constant over \(2 \pi, c\) is the speed of \(h\) ght, and \(k_{B}\) is Boltzmann's constant. a) Show that the total rate at which energy is radiated by a black body per unit area, over all frequencies, is $$ W=\frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x . $$ b) Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is. c) Even before Planck gave his theory of thermal radiation around the turn of the 20 th century, it was known that the total energy \(W\) given off by a black body per unit area per second followed Stefan's law: \(W=\sigma T^{4}\), where \(\sigma\) is the StefanBoltzmann constant. Use your value for the integral above to compute a value for the Stefan-Boltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.

Diffraction gratings: Light with wavelength \(\lambda\) is incident on a diffraction grating of total width \(w\), gets diffracted, is focused with a lens of focal length \(f\), and falls on a screen: Theory tells us that the intensity of the diffraction pattern on the screen, a distance \(x\) from the central axis of the system, is given by $$ I(x)=\left|\int_{-w / 2}^{w / 2} \sqrt{q(u)} \mathrm{e}^{i 2 \pi x u / \lambda f} \mathrm{~d} u\right|^{2} $$ where \(q(u)\) is the intensity transmission function of the diffraction grating at a distance \(u\) from the central axis, i.e., the fraction of the incident light that the grating lets through. a) Consider a grating with transmission function \(q(u)=\sin ^{2} \alpha u\). What is the separation of the "slits" in this grating, expressed in terms of \(\alpha\) ? b) Write a Python function \(q(u)\) that returns the transmission function \(q(u)=\sin ^{2} \alpha u\) as above at position \(u\) for a grating whose slits have separation \(20 \mu \mathrm{m}\). c) Use your function in a program to calculate and graph the intensity of the diffraction pattern produced by such a grating having ten slits in total, if the incident light has wavelength \(\lambda=500 \mathrm{~nm}\). Assume the lens has a focal length of 1 meter and the screen is \(10 \mathrm{~cm}\) wide. You can use whatever method you think appropriate for doing the integral. Once you've made your choice you'll also need to decide the number of sample points you'll use. What criteria play into this decision? Notice that the integrand in the equation for \(I(x)\) is complex, so you will have to use complex variables in your program. As mentioned in Section 2.2.5, there is a version of the math package for use with complex variables called cmath. In particular you may find the exp function from cmath useful because it can calculate the exponentials of complex arguments. d) Create a visualization of how the diffraction pattern would look on the screen using a density plot (see Section 3.3). Your plot should look something like this: e) Modify your program further to make pictures of the diffraction patterns produced by gratings with the following profiles: i) A transmission profile that obeys \(q(u)=\sin ^{2} \alpha d \sin ^{2} \beta u\), with \(\alpha\) as before and the same total grating width \(w\), and \(\beta=\frac{1}{2} \alpha\). ii) Two "square" slits, meaning slits with \(100 \%\) transmission through the slit and \(0 \%\) transmission everywhere else. Calculate the diffraction pattern for non-identical slits, one \(10 \mu \mathrm{m}\) wide and the other \(20 \mu \mathrm{m}\) wide, with a \(60 \mu \mathrm{m}\) gap between the two.
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