Explanation Given: The density of the material is ρ . Show the intersection of body at the arbitrary point ( x , y , z ) as in Figure (1). Conclusion: From Figure (1), Express the radius of the disk element. r = y (I) Here, the centroid of differential element along the y -axis is y . Compute the centroid of the differential element along the y -axis. z = h a 2 ( a 2 − y 2 ) y = a 1 − z / h Substitute a 1 − z / h for y in Equation (I). r = y = a 1 − z / h Express the mass of the disk element. d m = ρ d V (II) Here, the density of the material is ρ , and the volume of the body is V . Substitute π r 2 d z for d V in Equation (II). d m = ρ π r 2 d z (III) Substitute a 1 − z / h for r in Equation (III). d m = ρ π ( a 1 − z / h ) 2 d z = ρ π a 2 ( 1 − z / h ) d z Calculate the mass moment of inertia of this element about the z- axis. d I z = 1 2 d m r 2 (IV) Substitute ρ π r 2 d z for d m in Equation (IV). d I z = 1 2 ( ρ π r 2 d z ) r 2 = 1 2 ρ π r 4 d z (V) Substitute a 1 − z / h for r in Equation (V). d I z = 1 2 ρ π [ a 1 − z / h ] 4 d z = ρ π a 4 2 ( 1 − z / h ) 2 d z Express the mass of the solid. m = ∫ 0 h / 2 d m (VI) Substitute ρ π a 2 ( 1 − z / h ) d z for d m in Equation (VI)

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 10.8, Problem 90P

To determine

The mass moment of inertia Iz of the solid formed by revolving the shaded area around the z-axis in terms of the mass m of the solid.

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Chapter 10.8, Problem 89P

Chapter 10.8, Problem 91P

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Airplanes A and B, flying at constant velocity and at the same altitude, are tracking the eye of hurricane C. The relative velocity of C with respect to A is 300 kph 65.0° South of West, and the relative velocity of C with respect to B is 375 kph 50.0° South of East. A 120.0 km B 1N 1. Determine the relative velocity of B with respect to A. A ground-based radar indicates that hurricane C is moving at a speed of 40.0 kph due north. 2. Determine the velocity of airplane A. 3. Determine the velocity of airplane B. Consider that at the start of the tracking expedition, the distance between the planes is 120.0 km and their initial positions are horizontally collinear. 4. Given the velocities obtained in items 2 and 3, should the pilots of planes A and B be concerned whether the planes will collide at any given time? Prove using pertinent calculations. (Hint: x = x + vt) 0

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Chapter 10 Solutions

Engineering Mechanics: Statics

Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Prob. 1P Ch. 10.3 - Prob. 2P Ch. 10.3 - Prob. 3P Ch. 10.3 - Prob. 4P Ch. 10.3 - Prob. 5P Ch. 10.3 - Prob. 6P

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The mass moment of inertia I z of the solid formed by revolving the shaded area around the z -axis in terms of the mass m of the solid.

The mass moment of inertia Iz of the solid formed by revolving the shaded area around the z-axis in terms of the mass m of the solid.

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Chapter 10 Solutions