Sep 7 - In a rotational motion a particle travels in a...

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Moment of Inertia I = mr 2 How much torque is required to start the merry-go- round spinning? In a rotational motion, a particle travels in a circle so we need to think in terms of its position in radians, mass in moments of inertia, velocity in radians/sec and and momentum as angular momentum

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Formulation of the Particle on a Ring Problem: Consider a particle of mass m constrained to move in a ring of radius r in the xy plane with constant potential energy V(r) =0 . The sign of J z can be either positive or negative, signifying either clockwise or counterclockwise rotation. This example shows positive J z Circumference = 2  r 0 ≤  is the azimuthal angle in radians Right hand rule applies
PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY PHYSICAL CHEMISTRY: THERMODYNAMICS, STRUCTURE, AND CHANGE 10E | PETER ATKINS | JULIO DE PAULA ©2014 W. H. FREEMAN AND COMPANY CHAPTER 8: FIGURE 8C.2 Boundary Conditions: To be a single-valued function, after 2 radians, the wavefunction must repeat itself! An integer number of wavelengths must fit in the circumference of the ring We call this a cyclic boundary condition Not single-valued! An acceptable wavefunction 0 ≤  is the azimuthal angle in radians

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Circumference of the circle = 2  r So by inspection we can conclude : n =2  r or =2  r/n n = 0,1,2,3… 8C.3 If we apply the deBroglie relationship to angular momentum J z We conclude that angular momentum about the z axis J z is quantized: [Section 8C.1] The quantum number m l = 0, ±1, ±2, ±3, …. And since angular momentum J z is proportional to energy, only certain values of energy are allowed: 8C.4 m l = 0, ±1, ±2, ±3, …. A quantum of angular momentum J z  =  ±  pr E = J 2 /2I deBroglie says: Energy is quantized! p = h/ =2 r/n
PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY PHYSICAL CHEMISTRY: THERMODYNAMICS, STRUCTURE, AND CHANGE 10E | PETER ATKINS | JULIO DE PAULA ©2014 W. H. FREEMAN AND COMPANY

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