A thin rod of length L is placed along the x axis with its center at the origin

Question

                    A thin rod of length L is placed along the x axis with its center at the origin as shown in the figure. The rod has a total mass M, which is distributed along                     its lengh according to the linear mass density p(x)                 

                    
                

                    p(x)=Aabsolute x^5                 

                    
                

                    where A is constant with units KgM^5                 

                    
                

                    Find the expression for A in terms of ML and any whole numbers A= ___________?                 

                    
                

                    Calculate Iy the moment of inertia of this rod about the y axis. Answer with the numerical coefficient of ML^2

A thin rod of length L is placed along the x axis with its center at the origin, as shown in the figure. The rod has a total mass M, which is distributed along its length according to the linear mass density p (x): rho (x) = A|x5| Here A is a constant with units of kg/m6. Find an expression for A in terms of M, L and any whole numbers. Calculate /y, the moment of inertia of this rod about the y axis. Answer with the numerical coefficent of ML2. Iy =

Explanation / Answer

a) we use that integral of p dx = M

so

integral of A |x^5| from - L/2 to L/2

= 2* integral of A x^5 from 0 to L/2

= 2* A x^6/6 form 0 to L/2

= A (L/2)^6/3 = M

A = 192 M/L^6

Iy = integral of p x^2 dx from - L/2 to L/2

= 2* integral of A x^7 dx from - L/2 to L/2

= 2*A (x^8/8) from 0 to L/2

= 2* 192 M/L^6*1/8*(L/2)^8
=3/16 M L^2

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