Consider a system of two identical pendula of mass m, each suspended on a massless spring whose spring constant is k. The length of each spring at equilibrium is . The two pendula are coupled by a spring with spring constant ' and equilibrium length L. The generalised coordinates (21,01} and {22,02} encode the displacements of the two pendula from their equilibrium positions. Gravity acts downwards in the vertical direction. Here's a picture of the set up. L 01 ци Z1 k' e 02 k 22 L+AL (a) Consider for now the case of k' = 0, i.e., no coupling between the two pendula. Write down: (i) The kinetic energy, and (ii) The potential energy for pendulum 1 in this system in terms of the generalised coordinates {01, 1} and the physical properties of the system {m, l, k, g}. [Hint 1: There are two forms of potential energy in this reduced system. Hint 2: In polar coordinates, ||2 + |y2||² + 21012./
Consider a system of two identical pendula of mass m, each suspended on a massless spring whose spring constant is k. The length of each spring at equilibrium is . The two pendula are coupled by a spring with spring constant ' and equilibrium length L. The generalised coordinates (21,01} and {22,02} encode the displacements of the two pendula from their equilibrium positions. Gravity acts downwards in the vertical direction. Here's a picture of the set up. L 01 ци Z1 k' e 02 k 22 L+AL (a) Consider for now the case of k' = 0, i.e., no coupling between the two pendula. Write down: (i) The kinetic energy, and (ii) The potential energy for pendulum 1 in this system in terms of the generalised coordinates {01, 1} and the physical properties of the system {m, l, k, g}. [Hint 1: There are two forms of potential energy in this reduced system. Hint 2: In polar coordinates, ||2 + |y2||² + 21012./
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:Consider a system of two identical pendula of mass m, each suspended on a massless spring whose
spring constant is k. The length of each spring at equilibrium is . The two pendula are coupled by
a spring with spring constant ' and equilibrium length L. The generalised coordinates (21,01} and
{22,02} encode the displacements of the two pendula from their equilibrium positions. Gravity acts
downwards in the vertical direction. Here's a picture of the set up.
L
01
ци
Z1
k'
e
02
k
22
L+AL
(a) Consider for now the case of k' = 0, i.e., no coupling between the two pendula. Write down:
(i) The kinetic energy, and
(ii) The potential energy
for pendulum 1 in this system in terms of the generalised coordinates {01, 1} and the physical
properties of the system {m, l, k, g}. [Hint 1: There are two forms of potential energy in this
reduced system. Hint 2: In polar coordinates, ||2 + |y2||² + 21012./
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