30 lines
1.5 KiB
Markdown
30 lines
1.5 KiB
Markdown
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PROBLEMS
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PROBLEM 1. Obtain expressions for the Cartesian components and the magnitude of the
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angular momentum of a particle in cylindrical co-ordinates r, , Z.
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SOLUTION. Mx = m(rz-zi) sin - -mrzo cos ,
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My = -m(rz-zi) cos -mrzo sin ,
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Mz = mr2
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M2 =
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PROBLEM 2. The same as Problem 1, but in spherical co-ordinates r, 0, o.
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SOLUTION. Mx = -mr2(8 sin + sin 0 cos 0 cos ),
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My = mr2(j cos - sin 0 cos 0 sin b),
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Mz = mr2sin20,
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M2 =
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PROBLEM 3. Which components of momentum P and angular momentum M are conserved
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in motion in the following fields?
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(a) the field of an infinite homogeneous plane, (b) that of an infinite homogeneous cylinder,
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(c) that of an infinite homogeneous prism, (d) that of two points, (e) that of an infinite homo-
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geneous half-plane, (f) that of a homogeneous cone, (g) that of a homogeneous circular torus,
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(h) that of an infinite homogeneous cylindrical helix.
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SOLUTION. (a) Px, Py, Mz (if the plane is the xy-plane), (b) M, Pz (if the axis of the
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cylinder is the z-axis), (c) P (if the edges of the prism are parallel to the z-axis),
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(d) Mz (if the line joining the points is the z-axis), (e) Py (if the edge of the half-
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plane is the y-axis), (f) Mz (if the axis of the cone is the z-axis), (g) Mz (if the axis
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of the torus is the z-axis), (h) the Lagrangian is unchanged by a rotation through an angle
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so about the axis of the helix (let this be the z-axis) together with a translation through a
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distance h86/2m along the axis (h being the pitch of the helix). Hence SL = 8z aL/dz+
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+80 0L/26 = = 0, so that I+hPz/2n = constant.
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