The most populated rotational energy level of a linear rotor. By following the method used in Justification, derive an expression for the most populated rotational level of a spherical rotor, given that its degeneracy is (2J + 1 )2
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The most populated rotational energy level of a linear rotor. By following the method used in Justification, derive an expression for the most populated rotational level of a spherical rotor, given that its degeneracy is (2J + 1 )2
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- A non-uniform disk of mass M and radius R has its mass distributed in such a way that the mass per unit area is a function of the radial distance r from the center of the disk: σ(r) = b r , where b is a constant to be determined. What is the rotational inertia of this disk about an axis through the center of mass and perpendicular to the plane of the disk? The area differential can be written as a ring of radius r and thickness dr: dA = 2πrdr.(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m/v|d. Show that if r is the position of the particle then the vector J =r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = w X ri and that the total angular momentum J is J = Σm₁ [r}w - (r; · w)r;]. (ii) Show further that the component of J along the axis of rotation can be written as Iw, where I, the moment of inertia of the collection about the axis or rotation, is given by 1 = Σm₁p². Interpret pi geometrically. (iii) Prove that the total kinetic energy of the particles is 1².• Problem 3.22 The potential energy between the atoms of an hydrogen molecule can be modelled by means of the Morse potential -2(r – ra) (r V (r) = V exp 2 еxp where Vo = 7 × 10 12 erg, ) = 8 x 10-º cm and a = 5 x 10-º em.
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