Explanation Show the area of the differential element parallel to the y axis as in Figure (1). Conclusion: From the Figure 1: Compute the area of the differential element. d A = ( r d θ ) d r Here, the area of differential element is d A , differential angle is d θ , and differential radius is dr . Express the moment of inertia of the shaded area about the y axis. I y = ∫ A x 2 d A (I) Refer to Figure (1), the value for x is r cos θ . Substitute r cos θ for y and ( r d θ ) d r for dA in Equation (I). I x = ∫ A ( r cos θ ) 2 ( r d θ ) d r (II) Conclusion: Use double integration and apply the limits from 0 to r 0 and − α / 2 to α / 2 in Equation (II). I y = ∫ − α / 2 α / 2 ∫ 0 r 0 ( r cos θ ) 2 ( r d θ ) d r = ∫ − α / 2 α / 2

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Question

Chapter 10.3, Problem 24P

To determine

The moment of inertia of the shaded area about the y axis.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.3, Problem 23P

Chapter 10.4, Problem 5FP

Students have asked these similar questions

Water is supplied at 150 ft³/s and 70 psi to a hydraulic turbine through a 3-ft inside-diameter inlet pipe as indicated in the figure below. The turbine discharge pipe has a 4.8-ft inside diameter. The static pressure at section (2), 10 ft below the turbine inlet, is 10 in. Hg vacuum. If the turbine develops 2400 hp, determine the rate of loss of available energy between sections (1) and (2). Section (1) P₁ =70psi Q=150ft³/s D₁ = 3 ft 10 ft Turbine power loss = i P₂ = 10 in. Hg vacuum D₂ =4.8ft Section (2) de hp

This problem studies the response of two single degree of freedom bridge systems shown in Figure 1 under three loading cases. The problem has two parts. Part A and Part B use the same loading cases but the system is modified. Assume the following three loading cases in both Part A and Part B: (a) Harmonic wind load acting on the bridge deck pw(t) = powsin(ωwt) with amplitude pow and forcing circular frequency ωw. (b) Harmonic displacement base excitation acting at the base of the bridge pier ug(t) = ugosin(ωgt) with amplitude ugo and displacement circular frequency ωg. (c) Rectangular pulse load acting on the bridge deck with amplitude pop and pulse duration td. Part A The system includes part of a bridge deck and a bridge pier shown in Figure 1(a). For each loading case find the symbolic expression of the peak shear force in the bridge pier assuming the following: • The bridge deck is rigid and it has a mass m. • The bridge deck is rigidly connected with the bridge pier (i.e.,…

specific speed P #2 Q.2. A Pelton wheel turbine of 1.9 m diameter works under a head of 50 m at 150 rpm. The buckets are exposed to water jet which delivers from a nozzle of 20 cm in diameter. Find the overall efficiency power produced by the wheel if the buckets deflects the jet through an angle of 163°. coefficient of velocity as 0.98 [50 Marks] ·licosply Y and no Take the

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

Knowledge Booster