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A pendulum is suspended from the cusp of a cycloid cut in a rigid support (see Figure). The path described by the pendulum bob is cycloidal and is giben by x = a(0 – sin ø), y = a(coso– 1) where he length of the pendulum is l=4a, and where o is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with a frequency w = Vg/1, independent of the amplitude.

A pendulum is suspended from the cusp of a cycloid cut in a rigid support (see Figure). The path described by the pendulum bob is cycloidal and is giben by x = a(0 – sin ø), y = a(coso– 1) where he length of the pendulum is l=4a, and where o is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with a frequency w = Vg/1, independent of the amplitude.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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2. A pendulum is suspended from the cusp of a cycloid cut in a rigid support (see Figure). The path described by the pendulum bob is cycloidal and is giben by x = a(@ – sin ø), y= a(coso-1) where he length of the pendulum is l=4a, and where ø is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with a frequency w = Vg/1, independent of the amplitude. y 2a 1, X m Suggestion: write the total energy and show that, with an appropriate choice of variable of u = 4a cos, the energy can be put into the standard form of a simple 2 harmonic oscillator.
Transcribed Image Text:2. A pendulum is suspended from the cusp of a cycloid cut in a rigid support (see Figure). The path described by the pendulum bob is cycloidal and is giben by x = a(@ – sin ø), y= a(coso-1) where he length of the pendulum is l=4a, and where ø is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with a frequency w = Vg/1, independent of the amplitude. y 2a 1, X m Suggestion: write the total energy and show that, with an appropriate choice of variable of u = 4a cos, the energy can be put into the standard form of a simple 2 harmonic oscillator.
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