It is a function of the mass of the rotating body and the distance of the body from the axis of rotation. The Moment of Inertia(I) is independent of the angular velocity of the body. Hence, the Kinetic Energy of a Rotating Body is given by half of the product of the Moment of Inertia and the angular velocity of the body. Now we know that the Kinetic Energy of a body is given by Its angular velocity is then given by ω = v/r then v = rω. Let us assume a body of Mass ‘m’ rotating with velocity v at a distance ‘r’ from the axis of rotation. The moment of inertia of an object varies with respect to different axes of rotation. The moment of inertia of an object depends on its mass and its mass distribution relative to the axis of rotation. The inertia of an object depends only on its mass. The moment of inertia is that property of an object which opposes the change of state of the object in rotational motion. It is that property of an object which opposes the change of state of the object in linear motion. I = (hb/36)(b 2 – b 1 b + b 1 2 ) Difference Between Moment of Inertia and Inertia S.No. Moment of Inertia of different objects is discussed below in this article L is the perpendicular distance between two axes. Let in the above figure, we have to find the moment of inertia of I O of the body passing through the point O and about the axis perpendicular to the plane, while the moment of inertia of the body passing through the center of mass C and about an axis parallel to the given axis is I C, then according to this theorem If I x, I y, and I z are the moments of inertia of the body about the axis OX, OY, and OZ axes respectively, then according to this theoremĪccording to this theorem, the moment of inertia of a body about a given axis is the sum of the moment of inertia about an axis passing through the center of mass of that body and the product of the square of the mass of the body and the perpendicular distance between the two axes. The third axis is OZ which is perpendicular to the plane of the body and passes through the point of intersection of the OX and OY axes. In the above figure, OX and OY are two axes in the plane of the body which are perpendicular to each other. The sum of the moment of inertia of a body about two mutually perpendicular axes situated in the plane of a body is equal to the moment of inertia of the body about the third axis which is perpendicular to the two axes and passes through their point of intersection. There are two types of theorems that are very important with respect to the Moment of Inertia: Thus, the Radius of the Gyration of a body about an axis is equal to the square root of the ratio of the body about that axis. Thus, the Radius of Gyration of a body is perpendicular to the axis of rotation whose square multiplied by the mass of that body gives the moment of inertia of that body about that axis. If the mass and radius of gyration of the body are M and K respectively, then the moment of inertia of a body is The Radius of Gyration of a body is defined as the perpendicular distance from the axis of rotation to the point of mass whose mass is equal to the mass of the whole body and the Moment of Inertia is equal to the actual moment of inertia of the object as it has been assumed that total mass of the body is concentrated there. Uniform Plate or Rectangular Parallelepiped This table discusses expressions for the moment of inertia for some symmetric objects along with their rotation axis: ![]() I = ∑m i r i 2 Moment Of Inertia Formula for Different Shapes ![]() For non-uniform objects, we calculate the moment of inertia by taking the sum of the product of individual point masses at each different radius for this the formula used is.For uniform objects, the moment of inertia is calculated by taking the product of its mass with the square of its distance from the axis of rotation (r 2 ). ![]() Several ways are used to calculate the moment of inertia of any rotating object. Moment of Inertia of any object depends on the following values: ![]() I represent moment of inertia of the body about the axis of rotationįrom the equation, we can say that the moment of inertia of a body about a fixed axis is equal to the sum of the product of the mass of each particle of that body and the square of its perpendicular distance from the fixed axis. Now the moment of inertia of the entire body about the axis of rotation AB will be equal to the sum of the moment of inertia of all the particles, so Moment of inertia of n th particle = m n ×r n 2 Moment of inertia of third particle = m 3 ×r 3 2
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