Author: Haroon Khalil
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HOOKE’S LAW
Hooke’s law states that stress and strain are proportional to each other under elastic limit. Originally, Hooke’s law specified that stress was proportional to strain but later Thomas Young introduced constant of proportionality which is known as Young’s modulus of elastic. Further, this name was superseded by modulus of elasticity. Figure 12.3 demonstrates the Hooke’s Law. Figure…
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INTRODUCTION TO STRESS AND STRAIN
There are certain behaviours of all materials under the influence of external force. Stress and strain are one of the measures to show these behaviours. Stress is a resistive force per unit area, which is developed internally to oppose the external force subjected to the material. Strain is a measure of deformation of the material…
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Mass Moment of Inertia of a Circular Cone
Consider an elemental plate at a distance x from apex of radius r and thickness dx as shown in Figure 11.25. Mass of elemental plate = ρπr2dx. Figure 11.25 Mass Moment of Inertia of Circular Cone about Its Axis of Rotation The moment of inertia of circular plate about normal axis through its centre is Moment of inertia of the elemental plate about the axis…
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Mass Moment of Inertia of Sphere
Consider an elemental plate of thickness dy at a distance y from the diametral axis as shown in Figure 11.24. Radius of this elemental circular plate x is given by, Figure 11.24 Mass Moment of Inertia of Sphere x2 = R2 − y2 Mass of the elemental plate, dm = ρπx2 dy = ρπ (R2 − y2)dy Moment of inertia of the plate element about axis But, mass of sphere, Mass moment of inertia of hemisphere…
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Mass Moment of Inertia of a Hollow Cylinder
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…
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Mass Moment of Inertia of a Circular Disc
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
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Mass Moment of Inertia of a Circular Ring
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…
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MASS MOMENT OF INERTIA
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…
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Moment of Inertia of Some Composite Sections
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…
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Moment of Inertia of a Circular Disc
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…