MOMENT OF INERTIA FREE

When the bus stopped, your upper body moved forward whereas your lower body did not move. What did you experience at this point? Yes. After a few minutes, you arrive at a bus stop and the bus stops. The tensor R FEis the configuration-dependent resistance matrix representing the hydrodynamic forces and torques due to the imposed shear field.Imagine you are on a bus right now. The tensors E ∞and Ω ∞are, respectively, the position independent but time-dependent, anti-symmetric parts of the velocity-gradient tensor. x α for translation where x a is the position vector of the αth particle and | E ∞ | = γ ˙is the magnitude of the shear rate. The term U ∞is the velocity of the bulk shear flow evaluated at the particle centre where U ∞= Ω ∞for rotation, U ∞= E ∞. The shear field enters viathe terms U ∞, R FEand E ∞which are as follows: 1.

Thus, this term merely collects together all the previously defined terms. it depends on the generalised configuration vector xwhich specifies the position and orientation of all the particles. The term R FU( x) is the configuration-dependent resistance matrix generalised to include both translational and rotational terms, i.e. This is a generalisation of the hydrodynamic interactions present in the previous Langevin equation to include the effects of the bulk shear flow. The contribution of this degree of freedom to the specific heat, which is approximately 1 when the vibrations are fully excited, therefore begins to decrease as the temperature increases further, approaching asymptotically the value 1 2 typical of a rotation.

MOMENT OF INERTIA FREE

As the energy of the vibrations increases, their amplitude increases and ultimately, at sufficiently high temperatures, the vibration becomes a free rotation. One of the normal vibrations of the molecule is a “torsional” vibration, in which one of the CH 3 groups is twisted relative to the other.

This molecule consists of two CH 3 groups at a certain distance apart and oriented in a certain way to each other. We may mention the possibility of a curious change from vibration to rotation, an instance of which is afforded by the ethane molecule C 2H 6. In consequence the specific heat of polyatomic gases may often be regarded as approximately constant over fairly wide intervals of temperature. As the temperature increases, the various normal modes successively contribute to the specific heat. The various frequencies ω α for a polyatomic molecule generally range over a very wide interval. In practice, however, this limit is not reached, since polyatomic molecules usually decompose at considerably lower temperatures. The distance a is just the coordinate x C of the oxygen atom laying on y′ axis it can be found according eq. One can find the MI relative the axis z′ as a sum of two MI of MP being at the distance d from the z′ axis I z′ = 2 m 1 d 2.

The required MI is I z,C = I z ′ – ma 2*, where a is a distance between two parallel z axes. To find the molecule MI we shall take advantage of the theorem on parallel axis (see (1.3.48)): I z′ = I z ,C + ma 2, where I z′ is the MI of the molecule relative axis z′, passing through an oxygen atom and parallel to the axis z. We direct an axis z upwards perpendicular to a molecule planes. The origin (intersection of three axes C) is superimposed with the molecule CM. Calculate first the MI I z of the water molecule relative to an axis z′ which passes through oxygen atom perpendicular to a molecule plane. A water molecule consists of three MPs (nuclei of atoms) with a total mass m = 2 m 1 + m 2, where m 1 and m 2 are masses of hydrogen and oxygen atoms, respectively. Solution: Let's arrange a molecule as it is represented in Figure E1.13.

Moment of inertia