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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 isl蝇

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=12m蝇脳r路蝇脳r.

The value of 蝇脳riis 蝇脳ri=蔚ijk蝇jrk.

Substitute the above value in the equation mentioned below.

T=12m蔚ijk蝇jrk蔚ilm蝇lrm

The value of 蔚ijk蔚ilmis 蔚ijk蔚ilm=未ji未km-未jm未kl.

Substitute the above value in the equation mentioned below.

T=12m蝇j未ji未km-未jm未klrkrm蝇lT=12m蝇j未jirkrk-rjrl蝇l

The tensor of inertia is lji=未jir2-rjri.

Where r2=rkrk.

Hence, T=12m蝇jlji蝇.

The result has been proven.

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