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Chapter 7: Problem 1

Write the pressure at any depth below the surface of a liquid as a dyadic. Find the force \(d \mathbf{F}\) on an element of area \(d \mathbf{S}\) of a submerged body. Integrate this force over the surface of the body to establish Archimedes' principle. Hint: Use Gauss' theorem for a dyadic $$ \oint d \mathrm{~S} \cdot \mathrm{T}=\int \mathrm{\nabla} \cdot \mathrm{T} d \mathrm{~s} $$ as an aid in evaluating the integrals.

Short Answer

Expert verified
Answer: Archimedes' principle.

Step by step solution

01

Write down the pressure at any depth below the surface

To start, we know that pressure at depth \(h\) below the surface in a fluid of density \(\rho\) and under constant acceleration \(g\) due to gravity is given by \(p = \rho g h\). The force acting on an elemental area \(d\mathbf{S}\) is due to pressure, and its direction is normal to the surface. So, we can write the pressure as a dyadic product: $$ \mathbf{T} = \rho g h \mathbf{n} \otimes \mathbf{I}, $$ where \(\mathbf{n}\) is a unit normal vector to the surface element, and \(\mathbf{I}\) is the identity matrix.
02

Find the force on an elemental area

The force on an elemental area \(d\mathbf{S}\) due to pressure is given by the product of the pressure dyadic and the elemental surface vector: $$ d\mathbf{F} = \mathbf{T} \cdot d\mathbf{S} = \rho g h \mathbf{n} \otimes \mathbf{I} \cdot d\mathbf{S}. $$
03

Integrate the force over the surface of the body

To find the total force on the submerged body, we need to integrate \(d\mathbf{F}\) over the entire surface. Using Gauss' theorem for a dyadic, we can convert the surface integral into a volume integral: $$ \oint d\mathbf{S}\cdot \mathbf{T} = \int \nabla \cdot \mathbf{T}\, dV. $$ Plugging in the expression for \(\mathbf{T}\), we have: $$ \oint d\mathbf{S}\cdot(\rho g h \mathbf{n} \otimes \mathbf{I})=\int \nabla \cdot (\rho g h \mathbf{n} \otimes \mathbf{I})\, dV. $$
04

Calculate the divergence of the pressure dyadic

We now need to compute the divergence of \(\mathbf{T}\), which we can do component-wise. Recall, \(\nabla \cdot (\mathbf{A} \otimes \mathbf{B}) = (\nabla \cdot \mathbf{A})\mathbf{B} + \mathbf{A} \times (\nabla \times \mathbf{B})\). Using this formula, we get: $$ \nabla \cdot (\rho g h \mathbf{n} \otimes \mathbf{I}) = (\nabla\cdot(\rho g h \mathbf{n}))\mathbf{I} + \rho g h \mathbf{n} \times (\nabla \times \mathbf{I}). $$ Since the curl of the identity (\(\mathbf{I}\)) matrix is zero, the last term vanishes. So we have: $$ \nabla \cdot (\rho g h \mathbf{n} \otimes \mathbf{I}) = (\nabla\cdot(\rho g h \mathbf{n}))\mathbf{I}. $$
05

Compute the total force using the divergence theorem

Now we can use the divergence theorem to compute the total force acting on the submerged body: $$ \mathbf{F} = \oint d \mathbf{S} \cdot \mathbf{T} = \int (\nabla\cdot(\rho g h \mathbf{n}))\mathbf{I} \, dV = -\rho g\mathbf{k} \int_V \rho dV, $$ where \(\mathbf{k}\) is the unit vector in the upward vertical direction. Finally, the total force is given by $$ \mathbf{F} = -\rho g V_\text{disp} \mathbf{k}, $$ where \(V_\text{disp}\) is the volume of liquid displaced by the submerged body. This is Archimedes' principle, which states that the buoyant force acting on a submerged body is equal to the weight of the liquid displaced by the body.

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Most popular questions from this chapter

Prove that the three components \(A_{x}=A_{y t}, A_{y}=A_{z_{1}}, A_{z} \equiv A_{x_{y}}\) of an antisymmetric dyadic transform as the components of a vector and hence enable A to be treated as a physical vector field when it is a function of \(x, y, z\), and \(t\).

A vector has one scalar invariant-its magnitude or length-when its components are transformed by a rotation of axes. Show that a symmetric dyadic \(S\) is characterized by three scalar invariants. \((a)\) its trace, \(I_{1}=S_{11}+S_{22}+S_{33} ;(b)\) the sum of its diagonal minors, $$ I_{2}=\left|\begin{array}{ll} S_{11} & S_{12} \\ S_{21} & S_{12} \end{array}\right|+\left|\begin{array}{ll} S_{22} & S_{18} \\ S_{32} & S_{33} \end{array}\right|+\left|\begin{array}{ll} S_{33} & S_{21} \\ S_{13} & S_{11} \end{array}\right| $$ (c) its delerminanl, \(I_{3}=\left|S_{i j}\right|\).

Show that $$ \lambda+2 \mu=\frac{Y(1-\sigma)}{(1+\sigma)(1-2 \sigma)}=B+\frac{4}{3} \mu . $$ Show that when an external body force per unit volume \(\mathbf{F}_{1 e}\) is present, the equation for the elastic displacement (7.5.5) becomes $$ (\lambda+\mu) \nabla \nabla \cdot e+\mu \nabla \cdot \nabla_{e}+\mathbf{F}_{1 e}=\rho_{\varphi} \frac{\partial^{2} e}{\partial l^{2}} $$ (This equation becomes an equation for elastic equilibrium when \(\varphi\) is not a function of time. Its solution \(\rho(x, y, z)\) must then satisfy specified conditions at the boundary of a body. The stress and strain at each point of the body can be calculated knowing \(\theta\). We thus have formulated the problem of elastic equilibrium in a fundamental way.)

If \(V\) is a physical vector field, show that the divergence of \(V, \nabla \cdot \mathbf{V}\), is an invariant scalar field. \(H i n l ;\) By performing an axis rotation show that $$ \frac{\partial V_{x}}{\partial x}+\frac{\partial V_{z}}{\partial y}+\frac{\partial V_{z}}{\partial z}=\frac{\partial V_{z}^{\prime}}{\partial x^{\prime}}+\frac{\partial V_{y}^{\prime}}{\partial y^{\prime}}+\frac{\partial V_{z}^{\prime}}{\partial z^{\prime}} $$ when each side is calculated at the same point in space. It is also instructive to apply the test for invariance to the two products \(\mathbf{V} \cdot \mathbf{W}\) and \(\mathbf{V} \times \mathbf{W}\), where \(\mathbf{V}\) and \(\mathbf{W}\) are both physical vectors.

Define the mean pressure in a stressed elastic medium to be \(P=-\frac{1}{3}\left(f_{x x}+f_{x v}+f_{x t}\right)\), an invariant of the stress dyadic. Show that any stress dyadic \(\boldsymbol{F}\) can then be written as the sum of a pure shear dyadic and a dyadic representing mean pressure.
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