Pendulum Consider a pendulum of length L that swings by the force of gravity only.For

Chapter 4, Problem 39

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Consider a pendulum of length L that swings by the force of gravity only.

                                     

For small values of \(\theta=\theta(t)\), the motion of the pendulum can be approximated by the differential equation \(\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \theta=0\) where g is the acceleration due to gravity.

(a) Verify that \(\left\{\sin \sqrt{\frac{g}{L}} t, \cos \sqrt{\frac{g}{L}} t\right\}\) is a set of linearly independent solutions of the differential equation.

(b) Find the general solution of the differential equation and show that it can be written in the form

\(\theta(t)=A \cos \left[\sqrt{\frac{g}{L}}(t+\phi)\right]\)

Text Transcription:

theta=theta(t)

d^2 theta/dt^2 +g/L theta=0

{sin sqrt g/L t, cos sqrt g/L t}

theta(t)=A cos [sqrt g/L (t+phi)]