Lecture 12 Mass Moment of Inertia PDF

Center of Gravity & Mass moment of inertia Prof. Dr. Moutaz Hegazy Department of Mechanical Engineering  Center of Gravity Center of Gravity. A body is composed of an infinite number of particles of differential size, and so if the body is located with...

Center of Gravity & Mass moment of inertia Prof. Dr. Moutaz Hegazy Department of Mechanical Engineering  Center of Gravity Center of Gravity. A body is composed of an infinite number of particles of differential size, and so if the body is located within a gravitational field, then each of these particles will have a weight dW. These weights will form a parallel force system, and the resultant of this system is the total weight of the body, which passes through a single point called the center of gravity, G*.  Center of Gravity The location of the center of gravity The total weight of the body is the sum of the weights of all its particles, that is 𝑊 = ෍ 𝐹𝑧 𝑊 = න 𝑑𝑊 The location of the center of gravity, measured from the y axis, is determined by equating the moment of W about the y axis to the sum of the moments of the weights of all its particles about this same axis. Therefore, 𝑀𝑅𝑦 = ෍ 𝑀𝑦 𝑊 𝑥ҧ = න 𝑥𝑑𝑊 ෤ ‫𝑊𝑑𝑥 ׬‬ ෤ 𝑥ҧ = 𝑊  Center of Gravity Similarly, the location of the center of gravity, measured from the x axis, is determined by equating the moment of W about the x axis to the sum of the moments of the weights of all its particles about this same axis. Therefore, 𝑀𝑅𝑥 = ෍ 𝑀𝑥 𝑊 𝑦ത = න 𝑦𝑑𝑊 ෤ ‫𝑊𝑑𝑦 ׬‬ ෤ 𝑦ത = 𝑊 This idea can be generalized to a three-dimensional body and perform a moment balance about all three axes to locate the mass center G. ‫𝑊𝑑𝑥 ׬‬ ෤ ‫𝑊𝑑𝑦 ׬‬ ෤ ‫𝑊𝑑𝑧 ׬‬ ǁ 𝑥ҧ = 𝑦ത = 𝑧ҧ = 𝑊 𝑊 𝑊 Example: Locate the center of mass of the shown wire, where the density of the wire 10kg/m and the density of the plate 10 kg/m2.  Mass moment of inertia Consider a small mass m mounted on a rod of negligible mass which can rotate freely about an axis AA’. If a couple “M” due to tangent force “F” is applied to the system, the rod and mass, assumed to be initially at rest, will start rotating about AA’. y Apply Newton’s 2nd law on the particle of mass m in tangent direction F at ෍ 𝐹𝑡 = 𝑚𝑎𝑡 𝑎𝑡 = 𝛼𝑟 P r x Multiply both side by “r”   𝐹𝑟 = 𝑟 ∗ ∆𝑚𝛼𝑟 𝑀 = ∆𝑚 𝑟2 𝛼 The product mr2 provides, therefore, a measure of the inertia of the system, when we try to set it in rotation motion. For this reason, the product mr2 is called the mass moment of inertia of m with respect to the axis AA’.  Mass moment of inertia Consider now a body of mass m which is to be rotated about an axis AA’. Dividing the body into elements of mass m1, m2, … etc. The body’s resistance to being rotated is measured by the sum r12m1 + r22 m2+ …, which defines the moment of inertia of the body with respect to the axis AA’. Increasing the number of elements, we find that the moment of inertia is equal, in the limit, to the integral I = න 𝑟 2 𝑑𝑚  Mass moment of inertia Radius of gyration The radius of gyration k of the body with respect to the axis AA’ is defined by; 𝐼 I = 𝑚 𝑘2 𝑘= 𝑚 The radius of gyration k represents, therefore, the distance at which the entire mass of the body should be concentrated if its moment of inertia with respect to AA’ is to remain unchanged. If SI units are used, the radius of gyration k is expressed in meters and the mass m in kilograms, and thus the unit used for the mass moment of inertia is kg.m2.  Mass moment of inertia The moment of inertia of a body with respect to a coordinate axis can easily be expressed in terms of the coordinates x, y, z of the element of mass dm. 𝐼𝑥 = න 𝑦 2 + 𝑧 2 𝑑𝑚 𝐼𝑦 = න 𝑧 2 + 𝑥 2 𝑑𝑚 I = න 𝑟 2 𝑑𝑚 𝐼𝑧 = න 𝑥 2 + 𝑦 2 𝑑𝑚  Mass moment of inertia Parallel axes theory 𝑥 = 𝑥ҧ + 𝑥′ y = 𝑦ത + 𝑦′ z = 𝑧ҧ + 𝑧′ 𝐼𝑥 = න 𝑦 2 + 𝑧 2 𝑑𝑚 2 2 𝐼𝑥 = න 𝑦ത + 𝑦′ + 𝑧ҧ + 𝑧′ 𝑑𝑚 𝐼𝑥 = න 𝑦′2 + 𝑧′2 𝑑𝑚 + 𝑦ത 2 + 𝑧ҧ 2 න 𝑑𝑚 𝐼𝑥 = 𝐼𝑥′ + 𝑚 𝑑𝑥 2 similarly; 𝐼𝑦 = 𝐼𝑦′ + 𝑚 𝑑𝑦 2 𝐼𝑧 = 𝐼𝑧′ + 𝑚 𝑑𝑧 2 Example: The pendulum consists of two slender rods AB, and AC which have a mass density 3 kg/m. The thin plate has a mass density of 12 kg/m2. Determine, the mass center of the pendulum with respect to the x-y coordinate shown the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through C the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through the center of mass. y x