![]() ![]() The three diagonal elements are called the principal components of the moment of inertia and the corresponding axes that lead to this diagonal form are the principal axes of the molecule. If the coordinates are reoriented in this way the moment of inertia tensor becomes diagonal, i.e. It is essentially a reorientation of the orthogonal axis system. This is a transformation that preserves the lengths of the unit vectors along each axis and their mutual orthogonality. The moment of inertia tensor is a symmetric matrix and it can therefore be diagonalised by an orthogonal transformation of the Cartesian axes. Given a unit vector k in the direction of the rotation axis the moment of inertia about this axis is given by The moment of inertia tensor permits the calculation of the moment of inertia about any axis relative to the orientation of the molecule passing through the centre of mass. The form of the tensor is proved in many good texts on mechanics, such as Landau and Lifschitz. In a coordinate system fixed with respect to the molecule the components of the moment of inertia tensor are defined by To express the properties of the moment of inertia mathematically it is necessary to express it as a matrix. The analysis above shows clearly that the moment of inertia about an axis depends on the orientation of the axis relative to the molecule. You will not understand this until you have covered matrices and matrix diagonalisation in the maths course. It is easy to prove this theorem, taking the centre of mass as the originīecause the z coordinates of all the atoms are zero. All axes pass through the centre of mass. The moment of inertia about the z axis is the sum of the moments of inertia about the other two axes. This theorem applies to planar molecules. Should be familiar from the tutorial on collisions as the reduced mass. Where r is the bond length, and atom 2 at Putting the origin at the centre of mass, atom 1 is at Suppose the z-axis is aligned along the bond direction and the atoms are at z 1 and z 2. We now calculate the moment of inertia of a diatomic molecule about an axis perpendicular to the bond (this is one of the principal components - see later). The consequence the natural moment of inertia of a molecule is about an axis passing through the centre of mass, and it is straightforard to calculate it for any other axis. The result follows because the two central sums are zero from the definition of the centre of mass. Where M is the mass of the molecule and d the distance of the axis from the centre of mass. The moment of inertia about any other axis perpendicular to this may be found to be Thus the moment of inertia is minimised if the axis passes through the centre of mass of the molecule. The moment of inertia may be minimised with respect to the position of the axis, for example Where r i is the distance of atom i from the axis of rotation.Ĭonsider a molecule rotating about an axis parallel to the z-axis with fixed x and y coordinates. In chemistry we are most interested in the rotation of molecules, which are essentially made up of point masses, giving it measures the inertial towards angular acceleration. Mass moment for rigid bodies: Here the body is thought of as sum of particles each having mass dm.The moment of inertia of a single particle rotating about a centre was introduced in the tutorial on circular motion The formula for a single particle of mass m is The mass moment of inertia is the measurement of the distribution of the mass of an object or body relative to a given axis. Metric units: mm 4, cm 4, m 4 Mass Moment of Inertia: Units to express it : Imperial units: inches 4 Now, for the above figure we have the axis 0 given and hence we can calculate the moment of area by summing together l 2 dA for all the given elements of area dA in the yellow region.įor a rectangular region the area moment of inertia ![]() The second moment of the area around a given axis is called the area moment of inertia ![]()
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