Explanation Given information: The curve equation is y 2 + z 2 = k x Calculation: Sketch the parabolic section as shown in figure 1. Write the given curve Equation as follows: r 2 = y 2 + z 2 = k x (1) Calculate the radius of the elemental disk at a distance of x from the origin: At x = h , r = a Substitute a for r and h for x in Equation (1). a 2 = k ( h ) k = a 2 h Substitute a 2 h for k in Equation (1). r 2 = ( a 2 h ) x Determine the mass of the strip as shown below: d m = ρ d V = ρ ( π ( a 2 h ) x d x ) = p π ( a 2 h ) x d x (2) Integrate Equation (2) with respect to x and apply the limits. m = ∫ 0 h d m = ρ π a 2 h ∫ 0 h x d x = ρ π a 2 h [ x 2 2 ] 0 h = ρ π a 2 h [ h 2 2 ] m = 1 2 ρ π a 2 h Find the mass moment of inertia and radius of gyration with respect to x axis of the paraboloid by using direct integration. Consider the moment of inertia of an area by integration is ( 1 2 r 2 ) d m I x = ∫ 0 h d I x = ∫ 0 h ( 1 2 r 2 ) d m (3) Substitute p π ( a 2 h ) x d x for d m in Equation (3)

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ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Chapter 9.5, Problem 9.124P

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Find the mass moment of inertia and radius of gyration with respect to x axis of the paraboloid by using direct integration.

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Chapter 9.5, Problem 9.123P

Chapter 9.5, Problem 9.125P

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Find the mass moment of inertia and radius of gyration with respect to x axis of the paraboloid by using direct integration.

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Find the mass moment of inertia and radius of gyration with respect to x axis of the paraboloid by using direct integration.

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