Calculate the moment of inertia of a uniform cuboid of mass m and of dimension (l ×b ×h) about t

Calculate the moment of inertia of a uniform cuboid of mass...

Key Concepts

Moment of Inertia

The moment of inertia is a fundamental property of a rotating body that quantifies its resistance to angular acceleration about a chosen axis. It depends on both the total mass of the object and the distribution of that mass relative to the axis of rotation. This concept is central in rotational dynamics and is analogous to the role of mass in linear motion.

Uniform Mass Distribution

This concept refers to a mass that is evenly spread throughout an object. For a uniformly distributed object, each infinitesimal element contributes equally relative to its volume, which simplifies the process of calculating the moment of inertia through integration. Uniformity ensures that symmetry can be exploited in the analysis.

Axis of Rotation (Center-of-Mass Axis)

An axis of rotation through the center of mass is significant because when an object rotates about this axis, the distribution of mass is symmetric relative to the axis, simplifying the calculations. This axis is a principal axis of rotation in many symmetric objects, making it a common choice in practical applications of rotational dynamics.

Integration in Moment of Inertia Calculations

Integration is used to sum up the contributions of all infinitesimal mass elements within the object to the total moment of inertia. For continuous bodies, this method is essential since it takes into account the varying distances of mass elements from the axis of rotation, leading to a precise formulation of the object’s rotational inertia.

Rigid Body Geometry

The geometry of a rigid body, such as its dimensions and shape, directly affects its moment of inertia. Understanding the role of geometry is critical because the distance of each infinitesimal mass element from the axis of rotation depends on the shape and size of the object, influencing how the mass is distributed relative to the axis.

Transcript

00:02 Hello everyone, we are going to understand this question here given in the question given mass of cuboid that is m and dimension of cuboid that is length into breadth into height this is the dimension of cuboid so volume mass density of cuboid volume mass density that is total mass upon volume of cuboid l cross b l into b into h or we can write l into b b into h this is the mass density of cuboid now taking the partial quoid having dimension d x, d, d, y and dz so mass of partial cuboid mass of partial cubeoid mass of partial cuboid mass of partial cuboid that is represented by dm which is equal to which is equal to mass density into x square plus y square into sorry here mass density is equal to sigma into length is d x, d y and d z length is d x, width is dx into d -y into d -z.

02:18 Here, sigma is m upon l into b into x into d -x into d -x into d -z.

02:28 This is the mass of partial volume, mass of partial cuoid.

02:36 So, moment of inertia of partial mass, moment of inertia of partial mass, moment of inertia of partial firesal quiboyed i di is equal to d m into x square plus y square sorry it is dm into r square here r is equal to dm into x square plus y square root and squares so we can write d m into x square plus y square root so we can write d m into x square plus y square x square plus y square so now substituting the value of dm that is m upon l b h into d x into d x into d z and here value of r is x square plus y square now taking the integration of both side now taking integration, now taking integration, so we can write, d -i, sorry, i is equal to here triple integration because we have to integrate with respect to x, y and z.

04:56 So, x square plus y square into dx into d -y into d -z.

05:06 And mass density was m upon l into b into h.

05:12 So we can write m upon l into b into x and integration limit is from x square plus y square x square into dx, d, d, y, d z, x square into d x, into d x, into d x, into d x, plus y square into d x, x into dy into tz now here integration limit if we will integrate with respect to x then integration limit is from minus l y2 to plus ly to l b into h with respect to x integration limit is from minus l y2 to plus l y2 to plus l y2 that is x square into d x and integration limit is from minus h by 2 to plus hy2 into dy into minus by2 to plus by2 and integration limit is for integration dz plus y square plus y square so here integration limit is from minus ly2 to plus ly2 dx minus hy2 to plus hy2 plus hy2 into dy and minus by2 to plus by2 and this is dz.

07:22 Now doing further calculation so we will get m upon l into b into h and here integration limit that will give l is means x q upon x q upon 3 x q upon 3 x q y into z and here this is limit is from minus l by 2 to plus l y 2 and limit of y is minus b y 2 plus b by 2 now let's calculate it directly so we can write it as x q upon 3 and after substituting the value of l we will get l cube upon 8 plus l q upon 8 into h .y2 plus hy2 into by2 plus by2 plus by2...

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