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. The moment of inertia of a single particle rotating about a centre was introduced in the tutorial on circular motion
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