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Chapter 10: Q8P (page 520)

Do Problem (4.8) in tensor notation and compare the result with your solution of (4.8).

Short Answer

Expert verified

Answer

The result has been proven.

Step by step solution

01

Given Information

The kinetic energy is T and the tensor inertia is $l_{ω}$

02

Definition of a tensor of inertia.

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

03

Prove the result.

The formula states that kinetic energy,

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

The value of ${(ω \times r)}_{i}$ is ${(ω \times r)}_{i} = ε_{i j k} ω_{j} r_{k}$ .

Substitute the above value in the equation mentioned below.

$T = \frac{1}{2} m ε_{i j k} ω_{j} r_{k} ε_{i l m} ω_{l} r_{m}$

The value of $ε_{i j k} ε_{i l m}$ is $ε_{i j k} ε_{i l m} = δ_{j i} δ_{k m} - δ_{j m} δ_{k l}$ .

Substitute the above value in the equation mentioned below.

$T = \frac{1}{2} m ω_{j} (δ_{j i} δ_{k m} - δ_{j m} δ_{k l}) r_{k} r_{m} ω_{l} T = \frac{1}{2} m ω_{j} (δ_{j i} r_{k} r_{k} - r_{j} r_{l}) ω_{l}$

The tensor of inertia is $l_{j i} = (δ_{j i} r^{2} - r_{j} r_{i})$ .

Where $r^{2} = r_{k} r_{k} .$

Hence, $T = \frac{1}{2} m ω_{j} l_{j i} ω$ .

The result has been proven.

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Most popular questions from this chapter

Let $x = u + v, y = v$ . Find $d s$ , the a vectors, and $d S^{2}$ for the u, v coordinate system and show that it is not an orthogonal system. Hint: Show that the vectors are not orthogonal, and that $d S^{2}$ contains du dv terms. Write the $g_{i j}$ matrix and observe that it is symmetric but not diagonal. Sketch the lines $u, v = c o n s t$ and observe that they are not perpendicular to each other.

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

Let $T_{j k m n}$ be the tensor in $(5 .8)$ . This is a $4^{t h}$ -rank tensor and so has $3^{4} = 81$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after $(5 .8)$ .

${\begin{cases} x = u - v \\ y = 2 \sqrt{u v} \end{cases}$

Evaluate:

$\begin{array}{l} (a) δ_{i j} δ_{j k} δ_{k m} δ_{i m} \\ (c) ϵ_{j k 2} ϵ_{k 2 j} \\ (e) ϵ_{23 i} ϵ_{2 i 3} \end{array}$ $\begin{array}{l} (b) ϵ_{i j k} δ_{j k} \\ (d) ϵ_{3 j k} ϵ_{k j 3} \\ (f) ϵ_{k 31} ϵ_{3 k 1} \end{array}$

See all solutions

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