3. Consider a thin rod (of width zero and uniformly distributed mass) pivoted freely at one end about the horizontal z-axis. The rod is free to swing in the xy plane (x horizontal, y vertically down). Its mass is m, and total length is l. Answer the following questions: (a) Find the moment of inertia I, for the rotation of the bar about its pivot. Use an integral along the length and write your answer in terms of m and l. (b) Write down the total angular momentum L, and the torque t, in cylindrical coordinates. Write down the equation of motion L, = t, explicitly. (c) Assuming the motion is confined to small angles, find the simplified equation of motion. Show the period of this compound pendulum. (“Compound pendulum" is used to mean any pendulum whose mass is distributed – as contrasted with a "simple pendulum," whose mass is concentrated at a single point on a massless arm.) (d) What is the length of the equivalent simple pendulum, that is, the pendulum with the same period?
3. Consider a thin rod (of width zero and uniformly distributed mass) pivoted freely at one end about the horizontal z-axis. The rod is free to swing in the xy plane (x horizontal, y vertically down). Its mass is m, and total length is l. Answer the following questions: (a) Find the moment of inertia I, for the rotation of the bar about its pivot. Use an integral along the length and write your answer in terms of m and l. (b) Write down the total angular momentum L, and the torque t, in cylindrical coordinates. Write down the equation of motion L, = t, explicitly. (c) Assuming the motion is confined to small angles, find the simplified equation of motion. Show the period of this compound pendulum. (“Compound pendulum" is used to mean any pendulum whose mass is distributed – as contrasted with a "simple pendulum," whose mass is concentrated at a single point on a massless arm.) (d) What is the length of the equivalent simple pendulum, that is, the pendulum with the same period?
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Transcribed Image Text:3. Consider a thin rod (of width zero and uniformly distributed mass) pivoted freely at one end
about the horizontal z-axis. The rod is free to swing in the xy plane (x horizontal, y vertically
down). Its mass is m, and total length is l. Answer the following questions:
(a) Find the moment of inertia I, for the rotation of the bar about its pivot. Use an integral along
the length and write your answer in terms of m and l.
(b) Write down the total angular momentum L, and the torque t, in cylindrical coordinates. Write
down the equation of motion L, = t, explicitly.
(c) Assuming the motion is confined to small angles, find the simplified equation of motion. Show
the period of this compound pendulum. (“Compound pendulum" is used to mean any
pendulum whose mass is distributed – as contrasted with a "simple pendulum," whose mass is
concentrated at a single point on a massless arm.)
(d) What is the length of the equivalent simple pendulum, that is, the pendulum with the same
period?
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