Chapter 10: Q7P (page 520)

In $(7 .9)$ we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.

Short Answer

Expert verified

the inertia matrix is mentioned below.

I = (\begin{matrix} m (r^{2} - {x_{1}}^{2}) & - m x_{1} x_{2} & - m x_{1} x_{3} \\ - m x_{2} x_{1} & m (r^{2} - {x_{2}}^{2}) & - m x_{2} x_{3} \\ - m x_{3} x_{1} & - m x_{3} x_{2} & m (r^{2} - {x_{3}}^{2}) \end{matrix})

Step by step solution

Given Information

The elements of the first row of the inertia matrix are mentioned below.

\begin{matrix} I_{11} = m (r^{2} - {x_{1}}^{2}) \\ I_{12} = - m x_{1} x_{2} \\ I_{13} = - m x_{1} x_{3} \end{matrix}

Definition of a tensor of inertia.

The tensor of inertia is a mathematical representation of how mass is distributed in a rigid body.

Find inertia matrix.

The elements of the first row of the inertia matrix are mentioned below.

$\begin{matrix} I_{11} = m (r^{2} - {x_{1}}^{2}) \\ I_{12} = - m x_{1} x_{2} \\ I_{13} = - m x_{1} x_{3} \end{matrix}$

The Formula states that $I_{n j} = m (δ_{n j} r^{2} - x_{n} x_{j})$ .

Hence, the inertia matrix is mentioned below.

$I = (\begin{matrix} m (r^{2} - {x_{1}}^{2}) & - m x_{1} x_{2} & - m x_{1} x_{3} \\ - m x_{2} x_{1} & m (r^{2} - {x_{2}}^{2}) & - m x_{2} x_{3} \\ - m x_{3} x_{1} & - m x_{3} x_{2} & m (r^{2} - {x_{3}}^{2}) \end{matrix})$

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In $(7 .9)$ we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.

Short Answer

Step by step solution

Given Information

Definition of a tensor of inertia.

Find inertia matrix.

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Most popular questions from this chapter

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In(7.9)we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.

Short Answer

Step by step solution

Given Information

Definition of a tensor of inertia.

Find inertia matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Electromagnetism

Further Mechanics and Thermal Physics

Force

Torque and Rotational Motion

Dynamics

Study anywhere. Anytime. Across all devices.

In $(7 .9)$ we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.