31 lines
1.3 KiB
Markdown
31 lines
1.3 KiB
Markdown
PROBLEMS
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PROBLEM 1. Determine the period of oscillations of a simple pendulum (a particle of mass
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m suspended by a string of length l in a gravitational field) as a function of the amplitude of
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the oscillations.
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SOLUTION. The energy of the pendulum is E = 1ml2j2-mgl cos = -mgl cos to, where
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o is the angle between the string and the vertical, and to the maximum value of . Calculating
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the period as the time required to go from = 0 to = Do, multiplied by four, we find
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-cos
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po)
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The substitution sin $ = sin 10/sin 100 converts this to T = /(l/g)K(sin 100), where
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1/75
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-
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is the complete elliptic integral of the first kind. For sin 100 22 100 < 1 (small oscillations),
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an expansion of the function K gives
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T =
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§12
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Determination of the potential energy
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27
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The first term corresponds to the familiar formula.
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PROBLEM 2. Determine the period of oscillation, as a function of the energy, when a
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particle of mass m moves in fields for which the potential energy is
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(a) U = Alx
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(b) U = Uo/cosh2ax, -U0 0, (c) U = Uotan2ax.
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SOLUTION. (a):
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T =
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By the substitution yn = u the integral is reduced to a beta function, which can be expressed
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in terms of gamma functions:
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The dependence of T on E is in accordance with the law of mechanical similarity (10.2),
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(10.3).
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(b) T = (7/a)V(2m/E).
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(c) T =(t/a)v[2m/(E+U0)]
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