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)

12th Edition

ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

See similar textbooks

Concept explainers

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…

Videos

Question

Chapter 9.6, Problem 9.178P

To determine

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.6, Problem 9.177P

Chapter 9.6, Problem 9.179P

Students have asked these similar questions

3. A 15% magnesium chloride solution is flowing through a 5-nom sch 40 commercial steel pipe at a rate of 325,000 lbm/h. The average temperature of the magnesium chloride solution as it flows through the pipe is 10°F. Determine the convective heat transfer coefficient inside the pipe.

2. Jojoba oil is flowing through a ¾-nom stainless steel pipe at a flow rate of 1,850 lbm/h. After the velocity profile in the pipe is fully developed, the oil enters a heater, as shown in Figure P5.7. The length of the heater section is 5 ft. The properties of the jojoba oil at the average temperature in the heater section are given in Table P5.7. Determine the convective heat transfer coefficient inside the heater section of the pipe. ¾ nom stainless steel pipe Heater section L=5ft Fig. P5.7 TABLE P5.7 Thermophysical Properties of Jojoba Oil at the Average Temperature in the Heater P (lbm/ft³) 68.671 (Btu/lbm-R) 0.30339 μ (lbm/ft-s) 0.012095 k (Btu/h-ft-°F) 0.077424

1. Water is flowing inside of a 3-std type K copper tube at a flow rate of 1.2 kg/s. The average temperature of the water is 50°C. Cold, dry air at a temperature of 5°C and atmospheric pressure flows outside of the tube in cross flow with a velocity of 85 m/s. Determine the UA product for this tube under clean conditions.

Chapter 9 Solutions

VECTOR MECHANIC

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

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 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.