Moments and Products of The slender rod in figure P4.29 has a mass density give

Question

Moments and Products of The slender rod in figure P4.29 has a mass density give by p=p (x/L)2 + p0 in which p0 and p1 are constants. The rod has length L. Find its moment of inertia about the line defined by x = 0, y = L/2

Explanation / Answer

Moment Of intertia about Z axis and apply parallel axes theorem we have p = p1(x/L)^2 + p0 consider a dx element at x distance away moment of inertia along z axes = dmx^2 dm = density*volume = Adx*(p) dI = Apx^2dx =>I(z) = A*p1/L^2 *x^5/5 +A*p0x From x = 0 to X =l I(z)= Ap1L^3/5+Ap0L Applying Parallel Axes theorem mass m of the body = int dm = A*(p1/L^2 *x^3/3 + p0x) form 0 --> L ==> m = Ap1L/3 +Ap0L I (y =L/2) = I(z) + mL^2/4 = Ap1L^3/5 +Ap0L + Ap1L/12 + Ap0L/4 =Ap1L^3/5 + L(Ap0 +Ap1/12 +Ap0/4)

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