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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

Do Problem 3 in spherical coordinates; compare the results with Problem 8.3.

What are the physical components of the gradient in polar coordinates? [See (9.1)].The partial derivatives in (10.5) are the covariant components of $\nabla u$ . What relationdo you deduce between physical and covariant components? Answer the samequestions for spherical coordinates, and for an orthogonal coordinate system withscale factors $h_{1}, h_{2}, h_{3}$ .

Parabolic.

In spherical coordinates $\nabla^{2} r, \nabla^{2} (r^{2}), \nabla^{2} (\frac{1}{r^{2}}), \nabla^{2} e^{i k r c o s θ}$ .

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

See all solutions

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