Explanation Given: The width of the paraboloid is 200 mm . The height of the paraboloid is 200 mm . The density of material, ρ = 5 Mg / m 3 . Show the intersection of paraboloid at the arbitrary point ( x , y ) as in Figure 1. From Figure 1, Calculate the differential mass of the paraboloid. d m = ρ d V (I) Here, the density of material is ρ , and the volume of the paraboloid is V . Substitute π y 2 d x for d V in Equation (I). d m = ρ π y 2 d x (II) Here, the centroid of the differential element along the y - axis is y 2 . Substitute 50 x for y 2 in Equation (II). d m = ρ π ( 50 x ) d x Compute the moment of inertia of the differential element parallel to the y -axis about the x -axis. d I x = 1 2 d m y 2 (III) Substitute ρ π ( 50 x ) d x for d m and 50 x for y 2 in Equation (III). d I x = 1 2 [ ρ π ( 50 x ) d x ] ( 50 x ) = ρ π 2 ( 2500 x 2 ) d x Calculate the entire mass of the paraboloid. m = ∫ 0 200 mm d m (IV) Substitute ρ π ( 50 x ) d x for d m in Equation (IV). m = ∫ 0 200 mm ρ π ( 50 x ) d x = 50 ρ π ∫ 0 200 mm x d x = 50 ρ π [ x 2 2 ] 0 200 mm = 25 ρ π ( 200 ) 2 = 10 6 ρ π Conclusion: Express the moment of inertia of the paraboloid about the x -axis

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

See similar textbooks

Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

Videos

Question

Chapter 10.8, Problem 86P

To determine

The radius of gyration of the paraboloid.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.8, Problem 85P

Chapter 10.8, Problem 87P

Students have asked these similar questions

1.21 A one-dimensional plane wall is exposed to convective and radiative conditions at x = 0. The ambient and sur- rounding temperatures are T = 20°C and Tur = 40°C, respectively. The convection heat transfer coefficient is h=20 W/m² K, and the absorptivity of the exposed sur- face is α=0.78. Determine the convective and radiative heat fluxes to the wall at x = 0 if the wall surface tem- perature is T, = 24°C. Assume the exposed wall surface is gray, and the surroundings are large.

1.12 You've experienced convection cooling if you've ever extended your hand out the window of a moving vehi- cle or into a flowing water stream. With the surface of your hand at a temperature of 30°C, determine the con- vection heat flux for (a) a vehicle speed of 40 km/h in air at -8°C with a convection coefficient of 40 W/m² K and (b) a velocity of 0.2 m/s in a water stream at 10°C with a convection coefficient of 900 W/m²K. Which condition would feel colder? Contrast these results with a heat flux of approximately 30 W/m² under normal room conditions.

PLEASE SOLVE STEP BY STEP WITHOUT ARTIFICIAL INTELLIGENCE OR CHATGPT SOLVE BY HAND STEP BY STEP

Chapter 10 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

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.

The radius of gyration of the paraboloid.

Solution Summary: The author shows the intersection of paraboloid at the arbitrary point (x,y).

Concept explainers

Videos

The radius of gyration of the paraboloid.

Want to see the full answer?

Chapter 10 Solutions