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 corresponding to the particle being vertically below O. (a) Use the formulae for acceleration in polar coordinates, a = († — rġ²)î+ (2rė + rö)ê to show that 0 = -/-sin 0, where T is the tension in the rod. Use the relation (0²) = 200 to deduce that 2g 1 0². = T mg cos 0 + ml ², = cos 0 + A, and (1) where A is a constant. If the particle is instantaneously at rest (ẻ = 0) when the rod is horizontal, find 8 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 = = rî+rėô (2)
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 corresponding to the particle being vertically below O. (a) Use the formulae for acceleration in polar coordinates, a = († — rġ²)î+ (2rė + rö)ê to show that 0 = -/-sin 0, where T is the tension in the rod. Use the relation (0²) = 200 to deduce that 2g 1 0². = T mg cos 0 + ml ², = cos 0 + A, and (1) where A is a constant. If the particle is instantaneously at rest (ẻ = 0) when the rod is horizontal, find 8 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 = = rî+rėô (2)
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Please provide some direction to complete the part (a) and (b)
Transcribed Image Text: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 = († — rġ²)î + (2řÖ + rö)ê
to show that
T = mg cos 0 + mlġ²,
Ö 8
0 = — sin 0,
1
where T is the tension in the rod. Use the relation (0²) = 200 to deduce that
2g
8²
and
(1)
= cos 0 + A,
1
where A is a constant. If the particle is instantaneously at rest ( = 0) when the rod is horizontal,
find 8 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+rėÔ
(2)
together with (1) to write the equations of motion in polar coordinates. Show that, under
the small-angle approximation sin ≈ 0, the particle's position satisfies the fundamental
equation of damped harmonic motion.
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