Classical Mechanics Systems of Particles and Hamiltonian Dynamics 2nd. Edition by Greiner, Walter | 9783642034336 | 3642034330 | Holooly

Mechanics Dynamics

Classical Mechanics Systems of Particles and Hamiltonian Dynamics

88 SOLVED PROBLEMS

Question: 15.10

Calculate the normal frequencies of a symmetric molecule ABA of triangular shape: ...

Verified Answer:

Conservation of the center of gravity here reads [...

Question: 15.9

Find the vibration frequencies of a linear three-atom symmetric molecule ABA. We assume that the potential energy of the molecule depends only on the distances AB and BA and the angle ABA. Write the Lagrangian of the molecule in appropriate coordinates (normal coordinates) where the Lagrangian has ...

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We describe the geometry of the molecule in the x,...

Question: 13.9

Theancient Chinese court astronomers were able to predict lunar and solar eclipses with great reliability. The fact that such eclipses arise only occasionally—while otherwise we have a full moon or a new moon—is caused by the inclination of the orbital plane of the earth–moon system from the ...

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R_{0}

is defined as the vector poin...

Question: 11.12

(a) Find the moment of inertia of a thin homogeneous bar of length L with respect to an axis perpendicular to the bar.(b) A homogeneous bar of length L and mass m is supported at the ends by identical springs (spring constant k). The bar is moved at one end by a small displacement a and then ...

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(a) If the bar is divided into small segments of l...

Question: 12.3

Find the kinetic energy of a homogeneous circular top (density e, mass m, height h, vertex angle 2α), (a) rolling on a plane, and (b) whose base circle rolls on a plane while its longitudinal axis is parallel to the plane and the vertex is fixed at a point. ...

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For the calculation of the tensor of inertia, we c...

Question: 9.5

Which trajectory of the mass of a mathematical pendulum yields a pendulum period that is independent of the amplitude? ...

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We consider Fig. 9.4. From energy conservation, we...

Question: 9.3

Find the transverse displacement of a vibrating string of length l with fixed endpoints if the string is initially in its rest position and has a velocity distribution g(x). ...

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We look for the solution of the boundary value pro...

Question: 8.3

Determine the eigenfrequencies of the system of three equal masses suspended between springs with the spring constant k, as is shown in Fig 8.7. Hint: Consider the solution method of the preceding Exercise 8.2 and Mathematical Supplement 8.4. ...

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From Fig. 8.7, we extract for the equations of mot...

Question: 7.5

Three mass points are fixed equidistantly on a string that is fixed at its endpoints. (a) Determine the eigenfrequencies of this system if the string tension T can be considered constant (this holds for small amplitudes). (b) Discuss the eigenvibrations of the system. Hint: Note Exercises 8.1 and ...

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(a) For the equations of motion of the system, one...

Question: 7.3

When solving the determinant equation (7.19), we have made a mathematical restriction for c by setting c = 2 cos Θ Show that for the cases (a) |c| = 2,(b) c <−2 he eigenvalue equation DN = 0 cannot be satisfied. Clarify that thereby the special choice of the constant c is justified. ...

Verified Answer:

(a)

D_{n} = cD_{n−1} −D_{n−2}, D...

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