Moment of Inertia of Continious Bodies

MOMENT OF INERTIA OF CONTINUOUS BODIES Before delving into the definition of moment of inertia, it's crucial to understand the following key points: 1) Moment of inertia serves as the rotational counterpart of mass. 2) It reflects the body's resistance to any form of rotational motion. 0 Now, let's define moment of inertia as the product of mass and the square of its distance from the axis of rotation. Among these factors, the most pivotal is the distribution of mass concerning the axis of rotation. We'll proceed to explore the moment of inertia of various solid shapes around the specified axis. m 10 Expressed in equation form, it's written as I = m * r ^ 2 In general, for a uniform body with a mass element dm and moment of inertia dl, the moment of inertia of the element can be represented as dl = r ^ 2 dm. Integrating this formula yields the moment of inertia of the entire body as follows: integrate 1 dl = r ^ 2 * integrate 1 dm 40 The moment of inertia of a body is influenced by several factors: 1) Axis of rotation 2) Shape and size of the body 3) Density of the material comprising the body 4) Distribution of mass relative to the axis of rotation Among these factors, the distribution of mass relative to the axis of rotation stands out as the most significant. Let's now delve into the moment of inertia of different solid shapes around the given axis. dm o' Moment of Inertia of a Uniform Rod: To determine the moment of inertia of a uniform rod with mass M and length L about an axis that passes through its center and is perpendicular to its length, we employ the following approach: Let's examine a small element of mass dm and length dx located at a distance x from the center of the rod, as illustrated in the diagram. 0 dm x = (- L)/2 x = L/2 ㅈ dx The uniform rod maintains a constant linear mass density across its entire length, and this density can be defined as follows: dm = lambda*dx dl = dm * x ^ 2 dl = lambda*dx * x ^ 2 we obtain: Upon integration over the range from I = M/L * integrate x ^ 2 dx from (- L)/2 to L/2 I= M L [ x^ 3 3 ] (- L)/2 ^ (L/2) = M/L * [(L ^ 3)/24 - ((- L ^ 3)/24)] dI = M/L * x ^ 2 * dx (- L)/2 * tO * L/2 lambda = M/L A Now, let's examine a different scenario of rotation is positioned at one end of the rod. Given the rod's uniformity, its linear mass density remains constant throughout. I = M/L * (L ^ 3)/12 = (M * L ^ 2)/12 8 I = m * r ^ 2 We express the mass element dm in terms of the linear mass density, and a thickness dx is located at a distance x from one end of the rod. dI = lambda*dx * x ^ 2 dl = dm * x ^ 2 dm = lambda*dx dI = M/L * x ^ 2 * dx lambda = M/L dm x = L 0 x dx By integrating dl over the range from 0 to L, we can derive the moment of inertia of a rod with the axis of rotation situated at one of its ends, as follows: Sdl =xdx I= M hbar [(x ^ 2)/3] 0 ^ L I = (M * L ^ 3)/3 Moment of Inertia of a Uniform Rod: The process to determine the moment of inertia of a uniform ring, possessing mass M and radius R, about its centroidal axis is outlined. as follows: The process of deriving: Consider R as the distance of the mass element dm from the axis of rotation. lambda = M/(2pi*R) , dl = R(d*theta) I mng =R^ 2 int dm I ring = R ^ 2 * M/(2pi) * integrate 1 dtheta from 0 to 2pi I ring = M * R ^ 2 I rtng = R ^ 2 * integrate M/(2pi) dtheta from 0 to 2pi dm = M/(2mR) * R(d*theta) dm = M/(2pi) * d*theta dm = lambda*dl 0 Rightarrow Moment of Inertia of a Uniform Disc: The method to determine the moment of inertia of a uniform disc, with mass M and radius R, around its centroidal axis is detailed below. To calculate the moment of inertia of the uniform disc, we break down the disc into multiple uniform rings. Let's consider one of these rings, with a radius r and thickness dr, as depicted in the diagram. dA = Area of the ring dm = sigma*dA = M/(n * R ^ 2) * 2pi*rdr 9