H2. A simple pendulum has a particle of mass m at the end of a light rod of length 7. The other end of the rod is attached to a fixed point O, at the origin of polar coordinates (r, 0). The particle is at position (r, 0) with 0 = 0 corresponding to the particle being vertically below O. (a) Use the formulae for acceleration in polar coordinates, a ¹ = († — rġ²)ƒ + (2řė + rö)ê to show that T = mg cos 0 + mlė², 0=- == sin 0, where T' is the tension in the rod. Use the relation (0²) = 200 to deduce that 0²= 2g 1 con cos 0 + A, and (1) where A is a constant. If the particle is instantaneously at rest ( = 0) when the rod is horizontal, find and T when the rod is vertical. How does the tension in the vertical position (which is also the maximal tension) depend on the rod's length /?
H2. A simple pendulum has a particle of mass m at the end of a light rod of length 7. The other end of the rod is attached to a fixed point O, at the origin of polar coordinates (r, 0). The particle is at position (r, 0) with 0 = 0 corresponding to the particle being vertically below O. (a) Use the formulae for acceleration in polar coordinates, a ¹ = († — rġ²)ƒ + (2řė + rö)ê to show that T = mg cos 0 + mlė², 0=- == sin 0, where T' is the tension in the rod. Use the relation (0²) = 200 to deduce that 0²= 2g 1 con cos 0 + A, and (1) where A is a constant. If the particle is instantaneously at rest ( = 0) when the rod is horizontal, find and T when the rod is vertical. How does the tension in the vertical position (which is also the maximal tension) depend on the rod's length /?
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Transcribed Image Text:H2. A simple pendulum has a particle of mass m at the end of a light rod of length 1. The
other end of the rod is attached to a fixed point O, at the origin of polar coordinates (r, 0). The
particle is at position (r, 0) with 0 = 0 corresponding to the particle being vertically below O.
(a) Use the formulae for acceleration in polar coordinates,
a = ( − rġ²) + (2rė + rö)ê
to show that
cos 0 + A,
8
T = mg cos 0 + mlė²,
where T' is the tension in the rod. Use the relation (0²) = 200 to deduce that
2g
². =
1
and 0 ==
sin
sin 0,
(1)
where A is a constant. If the particle is instantaneously at rest ( = 0) when the rod is horizontal,
find and T when the rod is vertical. How does the tension in the vertical position (which is
also the maximal tension) depend on the rod's length /?
(b) Assume the particle is subject to linear air resistance av. Use the expression of the
velocity in polar coordinates
v=rf+rẻÔ
(2)
together with (1) to write the equations of motion in polar coordinates. Show that, under
the small-angle approximation sin 0 ≈ 0, the particle's position satisfies the fundamental
equation of damped harmonic motion.
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