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Chapter 10: Q5P (page 508)

Point masses 1 at (1, 1, 1) and at (-1, 1, 1).

Short Answer

Expert verified

Answer:

Inertia tensor is $t = (\begin{matrix} 4 & 0 & 0 \\ 0 & 4 & - 2 \\ 0 & - 2 & 4 \end{matrix})$ .

Step by step solution

01

Given Information

The given information is mentioned below:

$m_{1} = m_{2} = 1 r_{1} = (1, 1, 1) r_{2} = (- 1, 1, 1)$

02

Definition of Inertia tensor.

Inertia tensor is the angular momentum of a rigid body determined by the product of its tensor and its rotation vector.

03

Find the eigenvalues.

Solve for eigenvalues.

$l_{i j} = (3 δ_{i j} - 1) + (3 δ_{i j} - {(- 1)}^{δ_{1 l} + δ_{1 l}}) l = (\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}) + (\begin{matrix} 2 & 1 & 1 \\ 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{matrix}) = (\begin{matrix} 4 & 0 & 0 \\ 0 & 4 & - 2 \\ 0 & - 2 & 4 \end{matrix})$

04

Find principal axes and corresponding moment of inertia.

Find the moment of inertia.

$V_{1} = (1, 0, 0) V_{2} = (0, - 1, 1) V_{3} = (0, 1, 1)$

Simplify Further

$l_{1} = 4 l_{2} = 6 l_{3} = 2$

Hence, the inertia tensor is, $t = (\begin{matrix} 4 & 0 & 0 \\ 0 & 4 & - 2 \\ 0 & - 2 & 4 \end{matrix})$ .

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Most popular questions from this chapter

Interpret the elements of the matrices in Chapter 3, Problems 11.18 to11.21, as components of stress tensors. In each case diagonalize the matrix and so find the principal axes of the stress (along which the stress is pure tension or compression). Describe the stress relative to these axes. (See Example 1.)

$X_{i j k l} A_{k l} = B_{i j}$ .

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates $\nabla . e_{r}, \nabla . e_{θ}, \nabla \times e_{r}, \nabla \times e_{θ} .$ .

Use the results of Problem 1to find the velocity and acceleration components in spherical coordinates. Find the velocity in two ways: starting with ds and starting with $s = r e_{r}$ .

$T = \frac{1}{2} m (ω \times r) \cdot (ω \times r)$

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