Explanation Given information: The sum of moment of inertia is denoted by I x + I y + I z . The axis of revolution is denoted by x . The mass of the material is denoted by m . The radius of the sphere is denoted by a . Calculation: Consider that the distance from the origin O to the element mass dm will be denoted by r . Show the expression for the sum of moment of inertia I x + I y + I z as follows: I x + I y + I z = ∫ ( y 2 + z 2 ) d m + ∫ ( z 2 + x 2 ) d m + ∫ ( x 2 + y 2 ) d m = 2 ∫ ( x 2 + y 2 + z 2 ) d m = 2 ∫ r 2 d m Show the mass of the homogenous body as shown in Figure 1. Refer Figure 1. The mass of areas 1 and 2 m 1 , m 2 are equal. Consider that the volume corresponding to mass 1 m 1 and mass 2 m 2 are denoted by V 1 and V 2 . Show the relation between the moment of inertia of the body and the sphere as follows: ( I x + I y + I z ) b o d y = { ( I x + I y + I z ) s p h e r e + ( I x + I y + I z ) V 1 − ( I x + I y + I z ) V 2 } = { ( I x + I y + I z ) s p h e r e + 2 ∫ m 1 r 2 V 1 − 2 ∫ m 2 r 2 V 2 } (1) The mass of the element 1 and 2 are equal. The radius of element 1 r 1 is greater than 2 r 2 . Use the condition as below. ∫ m 1 r 2 d m − ∫ m 2 r 2 d m ≥ 0 (2) Modify Equation (1) using Equation (2)

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Author: BEER

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Moment of Inertia

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 9.178P

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Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O.

Prove that the mass moment of area Iy about the transverse axis y cannot be smaller than 3ma210.

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Chapter 9.6, Problem 9.177P

Chapter 9.6, Problem 9.179P

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Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O . Prove that the mass moment of area I y about the transverse axis y cannot be smaller than 3 m a 2 10 .

Solution Summary: The author demonstrates that the mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O.

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Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O.

Prove that the mass moment of area Iy about the transverse axis y cannot be smaller than 3ma210.

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Chapter 9 Solutions

Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O . Prove that the mass moment of area I y about the transverse axis y cannot be smaller than 3 m a 2 10 .

Solution Summary: The author demonstrates that the mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O.

Concept explainers

Videos

Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O. Prove that the mass moment of area Iy about the transverse axis y cannot be smaller than 3ma210.

Want to see the full answer?

Chapter 9 Solutions

Prove that the sum of mass moment of inertia of the body cannot be smaller than the similar sum computed for a sphere of same mass and same material centered at O.

Prove that the mass moment of area Iy about the transverse axis y cannot be smaller than 3ma210.