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Moment of Inertia

Tags: inertia axis mass

The moment of inertia (I) of a body is a measure of its rotational inertia, that is, its resistance to change in its angular velocity. It depends on the location of the axis, that is, on how the mass of the body is distributed relative to the axis. Thus, a body does not posses a unique moment of inertia; different axes through the body are associated with different moments of inertia.

For a discrete system of particles the moment of inertia is defined as

 I = ∑miri2           (15)

where  miis the mass of the ithparticle

and riis the perpendicular distance of the ithparticle from the axis of rotation.

 rotational kinematics
For a continuous system, the mass element should be chosen such that all the particles on it are at the same perpendicular distance from the axis. The moment of inertia of the whole body is given by

I =  ∫ r²dm                 (16)

where r is the perpendicular distance of the element dm from the axis.

angular velocity

 

 Illustration4

            Many molecules have a simple diatomic, dumbbell structure. Let us find the moments of inertia about four axes. We treat the bodies as point particles with mass m1 = 3 kg and m2 = 5 kg. Take d1 = 1m and d2 = 2 m in figure (7).

 angular speed

Solution

            rotational kinematics

Illustration5

Four point masses lie at the corners of a rectangle with sides of length 3 m and 4m, as shown in figure. Find the moment of inertia about each of the diagonals. Take M = 1 kg.  rotational kinematics equations

Solution

For each mass we need its perpendicular distance from the axis. For each axis, two masses do not contribute to the moment of inertia. The other two are at the same distance

                        3 sin53o = 2.4 m.

                        IA  = (4 kg)(2.4 m)2 + (2 kg)(2.4 m)2 = 34.6 kg m2

                        IB­ = (1 kg)(2.4 m)2 + (3 kg)(2.4 m2) = 23. 0 kg m2

Illustration 6

Find the moment of inertia of a thin rod of mass M and length L about an axis at one end and perpendicular to the rod, as shown in the figure.

 rotational kinematics problems and solutions

Solution

            The mass of an element of length dx is dm = λ dx,

            where λ = M/L is the linear mass density. 

            The moment of inertia of the mass element that is at a distance x from the axis is

rotational kinematics examples

The Parallel Axis Theorem

Let the moment of inertia of an object through its center of mass be Icm. The moment of inertia of the mass through any other axis parallel to the first is given by

                        Iparallel= Icm+ Mh2                                                        (17)

            where M is the total mass of the object and h is the perpendicular distance between the two parallel axes.

Perpendicular Axis Theorem

By a laminar body we mean an object that can be treated as two – dimensional, such as a large figure cut from a thin sheet of plywood. Let the z axis be perpendicular to a laminar body and run through any point O in the plane of the body. Let the moment of inertia of the body about that axis be Iz. Let the x and y be two mutually perpendicular axes lying in the plane of the body with the origin at the  point O as shown in figure. Let the moment of inertia about the x and y axes to be Ixand Iy. The perpendicular axis theorem states that

    Iz= Ix+ Iy          (18)

 Physics

 

Illustration7

Find the moment of inertia of a circular disc or solid cylinder of radius R about the following axes.

      (a)  through the center and perpendicular to the flat surface.

      (b)  at the rim and perpendicular to the flat surface

 

Solution

      (a)  Figure (11 a) shows that the appropriate mass element is a circular ring of radius  r and width dr. Its area is dA = 2πrdr and its mass is
dm = σdA, where σ = M/A is the areal mass density. The moment of inertia of this element is

rotational motion

 Torque

      (b)  The evaluation of the moment of inertia about an axis at the rim, as in figure, by integration is difficult. The parallel axis theorem, equation, with h = R, provides the answer with remarkable simplicity.

rotational motion definition

Table 2 Moment of Inertia of some common shaped Bodies

 rotational motion problems  rigid body
 Constant Angular Acceleration Model  32
.33  34
 35  36
 37  equations of rotation kinematics
 Linear Kinematics  angular displacement
 rotational kinematics experiment  average angular velocity

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