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Chapter 7: Problem 10

A finite stressed body with angular momentum \(h\) is subject to external surface forces which constitute a couple \(\mathbf{m}\). By use of the three- dimensional Gauss divergence theorem, convert the volume integral for \(\mathrm{dh} / \mathrm{d} t\) into an integral of the stresses \(t^{i j}\) over the surface of the body, and recognize this as \(\mathbf{m}\). [Hint: fili in the details of the following proof, where equation numbers indicate results used. $$ $$ \begin{aligned} &=\int\left[\cdots+t^{32}-t^{23}\right] \mathrm{d} V \\ &=\int\left[\left(x_{2} t^{3 j}\right)_{, j}-\left(x_{3} t^{2 j}\right)_{, j}\right] \mathrm{d} V \\ &=\oint\left(x_{2} t^{3 j} n_{j}-x_{3} t^{2 f} n_{j}\right) \mathrm{d} S=m_{1} \end{aligned} $$ and similarly for \(m_{2}\) and \(\left.m_{3} \cdot\right]\) \mathrm{d} h_{1} / \mathrm{d} t=\int\left[x_{2} t^{3 j}, j-x_{3} t^{2 j}, j+u_{2} g_{3}-u_{3} g_{2}\right] \mathrm{d} V $$

Short Answer

Expert verified
Question: Express the time derivative of angular momentum in terms of the surface stress integral and explain the steps involved in finding this expression. Answer: The time derivative of angular momentum, 𝑑ℎ/𝑑𝑡, can be expressed as: \(\frac{dh}{dt} = \oint \left[(x_{2}t^{3j}n_{j}) - (x_{3}t^{2j}n_{j}) + u_{2}g_{3} - u_{3}g_{2}\right] dS\) The steps involved in finding this expression are: 1. Write down the angular momentum of the body. 2. Apply the Gauss divergence theorem to convert the volume integral into a surface integral. 3. Recognize the surface integral as the components of the couple 𝐦. 4. Write the final expression for 𝑑ℎ/𝑑𝑡, showing it in terms of the surface stress integral.

Step by step solution

01

Write down the angular momentum of the body

Angular momentum, 𝑑ℎ/𝑑𝑡, can be written as: \(\frac{dh}{dt} = \int \left[\frac{\partial}{\partial t}(x_{2}t^{3j}) - \frac{\partial}{\partial t}(x_{3}t^{2j}) + u_{2}g_{3} - u_{3}g_{2}\right] dV\)
02

Apply the Gauss divergence theorem

The Gauss divergence theorem states that the volume integral of the divergence of a vector field is equal to the surface integral of that vector field: \(\int (\nabla \cdot \mathbf{V}) dV = \oint \mathbf{V} \cdot \mathbf{n} dS\) We apply this theorem to the volume integral given in step 1 to convert it into a surface integral. \(\frac{dh}{dt} = \oint \left[\frac{\partial}{\partial t}(x_{2}t^{3j}) - \frac{\partial}{\partial t}(x_{3}t^{2j}) + u_{2}g_{3} - u_{3}g_{2}\right] \cdot n_{j} dS\)
03

Recognize the result as the couple 𝐦

Now, we recognize the surface integral as the components of the couple 𝐦: \(m_{1} = \oint \left[(x_{2}t^{3j}n_{j})-(x_{3}t^{2j}n_{j})\right] dS\) And similarly for \(m_{2}\) and \(m_{3}\).
04

Write the final expression for 𝑑ℎ/𝑑𝑡

With the couple 𝐦 in terms of integrals over the surface of the body, the time derivative of the angular momentum can be expressed as: \(\frac{dh}{dt} = \oint \left[(x_{2}t^{3j}n_{j}) - (x_{3}t^{2j}n_{j}) + u_{2}g_{3} - u_{3}g_{2}\right] dS\) This expression shows how the volume integral for 𝑑ℎ/𝑑𝑡 has been converted into an integral of the stresses \(t^{ij}\) over the surface of the body, recognized as the couple 𝐦.

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Key Concepts

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

Angular Momentum in Special Relativity
Angular momentum is a fundamental concept in physics, which describes the quantity of rotation of an object and is an important feature in the study of special relativity. In a relativistic context, angular momentum is often discussed in terms of a stress-energy tensor, which includes not only the distribution of mass but also the distribution of energy and momentum.

Special relativity provides corrections for classical mechanics formulas for angular momentum, especially at high velocities, close to the speed of light. The conservation of angular momentum, which is a core principle in physics, implies that the total angular momentum of a closed system free of external torques remains constant over time. In our textbook example, the relationship between angular momentum (\(h\)) and moving surface forces gives a snapshot of how dynamic systems can be analyzed using the concepts of special relativity.
Gauss Divergence Theorem
The Gauss divergence theorem is a fundamental principle used in vector calculus and physics. The theorem relates the flow (divergence) of a vector field through a volume to the behavior of the vector field on the boundary of the volume. Mathematically, it connects the volume integral of the divergence of a vector field over a region to the surface integral of the vector field across the boundary of the region.

This theorem is particularly helpful in electromagnetism, fluid dynamics, and, as our example shows, in analyzing mechanical stress within a body. Applying the Gauss divergence theorem simplifies complex volume calculations into more manageable surface integrals, making it easier to solve physical problems involving vector fields.
Volume Integral
A volume integral in mathematics represents the integration of a function over a three-dimensional region. It is used in various physics applications to determine quantities that are distributed throughout a volume, such as mass, charge, or in our case, the rate of change of angular momentum under stress.

When solving problems involving volume integrals, the Gauss divergence theorem is often utilized to convert these into surface integrals. This conversion is incredibly useful as it facilitates the evaluation of complex volume integrals by reducing them to the integral over the surface that encloses the volume.
Surface Forces
Surface forces are forces that act over the surface of an object, as opposed to body forces like gravity which act throughout the volume of an object. In our exercise, the surface forces are responsible for creating a couple, or a pair of forces, which produces a torque on the body.

These forces are represented by the stress tensor when applied to a stressed body, as the tensor describes the distribution and magnitude of the forces that act on the surfaces of the body. Understanding how these surface forces interact and result in motion is crucial for designing structures and understanding material behavior under stress.
Stress Tensor
The stress tensor is a mathematical construct that represents the stress (force per unit area) within a material at a point. It's a second-order tensor, usually denoted as \( t^{ij} \), and describes the state of stress in a material body. The stress tensor is vital in determining how a material will deform in response to internal or external forces and is critical for engineering, materials science, and physics.

In our problem, the stress tensor provides the means to convert volume integrals into surface integrals through the Gauss divergence theorem, ultimately allowing us to solve for the angular momentum change in response to the external forces applied to the surface of the body.

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Most popular questions from this chapter

Consider Lewis and Tolman's lever paradox. The pivot B of a right-angled lever \(A B C\) is fixed at the origin of \(S^{\prime}\) while \(A\) and \(C\) lie on the positive \(x\) and \(y\)-axes, respectively, and \(\mathrm{AB}=\mathrm{BC}=a\). Two numerically equal forces of magnitude \(f\) act at \(A\) and \(C\), in directions \(B C\) and \(B A\), respectively, so that equilibrium obtains. Show that in the usual second frame S there is a clock wise couple \(f a v^{2} / c^{2}\) duc to these forces. Why does the lever not turn? Resolve the paradox along the lines of the preceding exercise. [Note: Intuitively it is not difficult to see how the angular momentum \(h\) of the lever continually increases in \(S\) in spite of its non-rotation. The force at \(C\) continually does positive work on the lever, while the reaction at B does equal negative work. Thus energy flows in at \(\mathrm{C}\) and out at B. But an energy current corresponds to a momentum density. So there is a constant nonmaterial momentum density along the limb \(\mathrm{BC}\) towards \(\mathrm{B}\). It is the continual increase of its moment about the origin of \(S\) that causes \(\mathrm{d} / \mathrm{d} t\) to be non- zero.]

From the form \((49.5)\) of its energy tensor and the cquation \(T^{\mu v} . v=0\) prove directly that every portion of an incoherent fluid, subject to no external forces, moves with uniform velocity, i.e. that \(A^{\mu}\) \(=\mathrm{d} U^{\mu} / \mathrm{d} \tau=0 .\left[H i n t:\right.\) expand \(\left\\{\left(\rho_{0} U^{\mu}\right) U^{\mu}\right\\}_{, \nu}=0\) and use \(U^{\mu},{ }_{, v} U^{v}\) \(\left.=A^{\mu}, U_{\mu} A^{\mu}=0 .\right]\)

Assume that, at some event, an energy tensor \(T^{\mu v}\) satisfies \(T^{\mu \nu} V_{\mu} V_{v}>0\) for all timelike and null vectors \(V^{\mu}\). By reference to (43.13) show that for an electromagnetic energy tensor this inequality does not necessarily hold. But if it holds, prove that there is at most one 'rest frame' for \(T^{\mu v}\), i.e. a frame in which \(T^{0 i}=0\). [Hint: the fourvelocity of the required frame satisfies \(T^{\mu v} U_{v}=k U^{\mu}\) for some nonzero \(k\) (why?); if there were two such timelike 'eigenvectors' of \(T^{\mu \nu}\) then there would also be a null eigenvector.] Note: one can also show that the inequality implies the existence of at least one rest frame, and, conversely, that a unique rest frame implies the inequality.

A steel cable of proper density \(\rho_{0}\) is subjected to a tension \(t\) per unit cross-sectional area. Find the maximum value of \(t\) for which \(\rho>0\) in all frames. (Actual steel cables break under a tension of about \(200 \mathrm{~kg}-\mathrm{wt} / \mathrm{mm}^{2}\).)

The analogue for fluids of the 'rigid motion' of solids is 'incompressible motion': this is defined to be such that the proper volume of each fluid element is conserved. Prove that the necessary and sufficient condition for such motion is \(U^{\mu}, \mu=0\). Prove also that, in the case of perfect fluids subject to pure external forces only, this condition is equivalent to \(\mathrm{d} \rho_{0} / \mathrm{d} t=0\), i.e. the constancy of \(\rho_{0}\) for each fluid element.
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