Q6P Point masses 1 at (1, 1, 聽-2) a... [FREE SOLUTION]

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Chapter 10: Q6P (page 508)

Point masses 1 at (1, 1, -2) and 2 at (1, 1, 1).

Short Answer

Expert verified

Answer:

Inertia tensor is $l = (\begin{matrix} 9 & - 3 & 0 \\ - 3 & 9 & 0 \\ 0 & 0 & 6 \end{matrix})$ .

Step by step solution

Given Information.

The given information is mentioned below:

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

Definition of Point mass.

The concept of a physical object with non-zero mass but explicitly and particularly minuscule volume or linear dimension is known as point mass.

Find the eigenvalues.

Solve for eigenvalues.

$l_{i j} = (6 未_{i j} - {(- 2)}^{未_{3 i} + 未_{3 i}}) + 2 (3 未_{i j} - 1) l = (\begin{matrix} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{matrix}) - (\begin{matrix} 1 & 1 & - 2 \\ 1 & 1 & - 2 \\ - 2 & - 2 & 4 \end{matrix}) + 2 (\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}) = (\begin{matrix} 5 & - 1 & 2 \\ - 1 & 5 & 2 \\ 2 & 2 & 2 \end{matrix}) + (\begin{matrix} 4 & - 2 & - 2 \\ - 2 & 4 & - 2 \\ - 2 & - 2 & 4 \end{matrix}) l = (\begin{matrix} 9 & - 3 & 0 \\ - 3 & 9 & 0 \\ 0 & 0 & 6 \end{matrix})$

Find principal axes and corresponding moment of inertia.

Find the moment of inertia.

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

Simplify further.

$l_{1} = 12 l_{2} = 6 l_{3} = 6$

Hence, inertia tensor is, $l = (\begin{matrix} 9 & - 3 & 0 \\ - 3 & 9 & 0 \\ 0 & 0 & 6 \end{matrix})$

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91影视

Point masses 1 at (1, 1, -2) and 2 at (1, 1, 1).

Short Answer

Step by step solution

Given Information.

Definition of Point mass.

Find the eigenvalues.

Find principal axes and corresponding moment of inertia.

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