Category: Advance Concept Mechanical Eng

Let in Figure 11.23, Ro = Outer radius of cylinder; Ri = Inner radius of the cylinder, L = Length of cylinder; ρ = Mass density of the cylinder; and M = Mass of the cylinder. Figure 11.23 Mass Moment of Inertia of a Hollow Cylinder Consider a small elemental ring of width dr at a distance of radius r from the centre of cylinder then the mass of the element, dm = ρ·2πr·dr·L…

Consider an element of the disc of arc length rdθ, width dr, and thickness t. ρ is the mass density of the disc. Mass moment of inertia of the disc can be calculated as shown in Figure 11.22. Figure 11.22 Mass Moment of Inertia of a Circular Disc Moment of inertia perpendicular to the plane of the circular disc

Consider a circular ring of radius R as shown in Figure 11.21. Let the mass per unit length of the ring is m. To find the mass moment of inertia of the ring about the diametral axis XX, consider an element of length ds = rdθ; the distance of the element from the diametral axis XX is R sin θ; and mass of the element is mrdθ. Figure…

Mass moment of inertia of a body about an axis is defined as the sum of the product of its elemental masses and square of their distances from the axis. Radius of gyration of a solid body can be given as where I is mass moment of inertia, M is the mass of the body, and k is the radius of…

The composite section in Figure 11.20 can be divided into three parts—triangular part of area A1, rectangular part of area A2, and semicircular part of area A3. The individual centroid for each section is shown in Figure 11.20 as C1, C2, and C3. The centroid of entire section is located at C. Consider the moment of inertia of the sections 1, 2, and 3 about the…

Consider an element of the arc length rdθ and width dr of the circular disc as shown in Figure 11.19. Now, the moment of inertia of the element about diametral axis x−x is given by, Figure 11.19 Moment of Inertia of a Circular Disc about Its Diametral Axis  IXX = y2 dA = (r sin θ)2 r dθ dr = r3 sin2 θ dθ dr   Now, moment of inertia of entire circle about diametral…

A. Moment of Inertia of a Rectangle Consider an elemental strip of width dy at distance of y from a centroidal axis of a rectangle as shown in Figure 11.17. Figure 11.17 Moment of Inertia of a Rectangle about Centroidal Axis Moment of inertia of the strip is given by,   IXX = y2 dA = y2bdy   Now, moment of inertia of entire rectangle can…

Theorem of parallel axis states that if the moment of inertia of a plane area about an axis in the plane of the area through centroid be represented by IG, then the moment of inertia of the plane about a parallel axis AB (IAB) in the plane at a distance h from the centroid of the area is…

Theorem of perpendicular axis states that if IXX and IYY be the moment of inertia of a plane section about two mutually perpendicular axes X−X and Y−Y in the plane of the section (as shown in Figure 11.15), then the moment of inertia of the section IZZ about the axis Z−Z, perpendicular to the plane and passing through the intersection of axes X−X and Y−Y is given by, Figure 11.15 Perpendicular Axis…

Radius of gyration of a body about an axis is a distance such that its square multiplied by the area gives moment of inertia of the area about the given axis.