(a) Explanation Given information: The mass of tetrahedron is m . Calculation : Sketch the tetrahedron series of vertical slices as shown in Figure 1. First divide the tetrahedron into a series of thin vertical slices of thickness d z as shown below: x = − a c z + a = a ( 1 − z c ) y = − b c z + b = b ( 1 − z c ) Find the mass d m of the slab as shown below: d m = ρ d V = ρ ( 1 2 x y d z ) = 1 2 ρ a b ( z c ) 2 d z Integrate the m as shown below: m = ∫ d m = ∫ 0 c 1 2 ρ a b ( 1 − z c ) 2 d z = 1 2 ρ a b [ − c 3 ( 1 − z c ) 3 ] 0 c = 1 2 ρ a b [ − c 3 ( 1 − c c ) 3 ] m = 1 6 ρ a b c Find the direct integration the mass product of inertia I z x by using the formula as shown below: d I z x = d I z ′ x ′ + z E L x E L d m (b) To determine Deduce I y z and I x y from the result obtained in part a .

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ISBN: 9781259977268

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek

Publisher: McGraw-Hill Education

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Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 9.6, Problem 9.162P

(a)

To determine

Find the mass product of inertia $Izx$ by direct integration.

(b)

To determine

Deduce $Iyz$ and $Ixy$ from the result obtained in part a.

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Chapter 9.6, Problem 9.161P

Chapter 9.6, Problem 9.163P

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Find the mass product of inertia I z x by direct integration.

Solution Summary: The author explains how to find the mass product of inertia I_zx by direct integration.

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Find the mass product of inertia $Izx$ by direct integration.

Deduce $Iyz$ and $Ixy$ from the result obtained in part a.

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Chapter 9 Solutions

Find the mass product of inertia I z x by direct integration.

Solution Summary: The author explains how to find the mass product of inertia I_zx by direct integration.

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Find the mass product of inertia Izx<m:math><m:mrow><m:msub><m:mi>I</m:mi><m:mrow><m:mi>z</m:mi><m:mi>x</m:mi></m:mrow></m:msub></m:mrow></m:math> by direct integration.

Deduce Iyz<m:math><m:mrow><m:msub><m:mi>I</m:mi><m:mrow><m:mi>y</m:mi><m:mi>z</m:mi></m:mrow></m:msub></m:mrow></m:math> and Ixy<m:math><m:mrow><m:msub><m:mi>I</m:mi><m:mrow><m:mi>x</m:mi><m:mi>y</m:mi></m:mrow></m:msub></m:mrow></m:math> from the result obtained in part a.

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Chapter 9 Solutions

Find the mass product of inertia $Izx$ by direct integration.

Deduce $Iyz$ and $Ixy$ from the result obtained in part a.