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2. 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 0. (a) Use the formulae for acceleration in polar coordinates, a = ( − rė²)ŕ+ (2ŕė +rë)ê (2) to show that T = mg cos 0+mlo², and Ö 0 = -²² sin 0, 3 where T is the tension in the rod. Use the relation (02) = 200 to deduce that j2 2g cos 0 + A, 1 where A is a constant. If the particle is instantaneously at rest (0 = 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 1? (b) Assume the particle is subject to linear air resistance -av. Use the expression of the velocity in polar coordinates v =ŕŕ+rð (3) together with (2) 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.

2. 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 0. (a) Use the formulae for acceleration in polar coordinates, a = ( − rė²)ŕ+ (2ŕė +rë)ê (2) to show that T = mg cos 0+mlo², and Ö 0 = -²² sin 0, 3 where T is the tension in the rod. Use the relation (02) = 200 to deduce that j2 2g cos 0 + A, 1 where A is a constant. If the particle is instantaneously at rest (0 = 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 1? (b) Assume the particle is subject to linear air resistance -av. Use the expression of the velocity in polar coordinates v =ŕŕ+rð (3) together with (2) 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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. 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 0. (a) Use the formulae for acceleration in polar coordinates, a = ( − rė²)ŕ+ (2ŕė +rë)ê (2) to show that T = mg cos 0+mlo², and Ö 0 = -²² sin 0, 3 where T is the tension in the rod. Use the relation (02) = 200 to deduce that j2 2g cos 0 + A, 1 where A is a constant. If the particle is instantaneously at rest (0 = 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 1? (b) Assume the particle is subject to linear air resistance -av. Use the expression of the velocity in polar coordinates v =ŕŕ+rð (3) together with (2) 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.
Transcribed Image Text:2. 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 0. (a) Use the formulae for acceleration in polar coordinates, a = ( − rė²)ŕ+ (2ŕė +rë)ê (2) to show that T = mg cos 0+mlo², and Ö 0 = -²² sin 0, 3 where T is the tension in the rod. Use the relation (02) = 200 to deduce that j2 2g cos 0 + A, 1 where A is a constant. If the particle is instantaneously at rest (0 = 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 1? (b) Assume the particle is subject to linear air resistance -av. Use the expression of the velocity in polar coordinates v =ŕŕ+rð (3) together with (2) 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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