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lecture2 - Lecture 2 Notes: 06 / 28 The Simple Pendulum:...

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Lecture 2 Notes: 06 / 28 The Simple Pendulum: Force Diagram A simple pendulum consists of a small mass suspended on an approximately massless, non-stretchable string. It is free to oscillate from side to side. The forces acting on the mass are the force of gravity and the tension in the string: The tension cancels out the component of mg that lies along the string; this keeps the object from accelerating in the direction of the string, and thus keeps the string's length constant. The net force is simply the remaining component of mg , which is pointed perpendicular to the string and is equal to: (We have made the net force negative since it points in the direction of decreasing θ ). Equations of Motion The mass travels along an arc on a circle; the displacement along the arc is given by x = L and so the acceleration is given by a = L α , where = d 2 θ /dt 2 . Thus, Newton's Law gives
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This is not Hooke's Law, since a sine function appears on the right hand side. However, for small angles, the sine function can be expanded as follows: The angle here must be expressed in radians. If the angle is sufficiently small, then we can just keep the first term in the expansion, and replace sin ( θ ) with . Our equation of motion then becomes This is clearly the equation of motion for a harmonic oscillator, with playing the role of x , α taking the role of a , and k = mg / L . Thus, for small displacements, the pendulum will oscillate with simple harmonic motion (this is just another example of simple harmonic motion being a nearly universal behavior for systems near equilibrium). Using the results from the previous lecture, the pendulum's angular frequency, frequency and period are: Note that the pendulum's frequency does not depend on how heavy an object is attached to the string: all masses oscillate with the same frequency. However, the pendulum is sensitive to the length of the string and the acceleration due to gravity. Energy of the Pendulum The pendulum only has gravitational potential energy, as gravity is the only force that does any work. Let us define the potential energy as being zero when the pendulum is at the bottom of the swing, = 0 . When the pendulum is elsewhere, its vertical displacement from the = 0 point is h = L - L cos ( ) (see diagram)
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The potential energy of the pendulum is therefore For small angles, this turns out to correspond to the potential energy of the harmonic oscillator, just as the force does. We can expand the cosine function as follows:
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lecture2 - Lecture 2 Notes: 06 / 28 The Simple Pendulum:...

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