Simple pendulum
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Simple pendulum
A simple pendulum is a bob or object with mass suspended by an inextensible string or rod from a point, where the connection is frictionless. [Diagram goes here - download the original to see it.] When the bob is displaced from the vertical, it will oscillate. In fact, these oscillations do not obey simple harmonic motion. One way to see intuitively that the simple pendulum does not in fact oscillate with simple harmonic motion is to realise that if the bob is given sufficient energy it will complete a whole revolution, and carry on indefinitely in one direction - assuming that the connection is frictionless, and the object is in a vacuum. [Diagram goes here - download the original to see it.] So as the pendulum system acquires more and more energy, its motion becomes less and less like a simple harmonic oscillator. Nevertheless, if the angle of the swing of the pendulum is small, a pendulum does behave approximately like a simple harmonic oscillator, which we will now show. [Diagram goes here - download the original to see it.] That is, we require the angle, x, in the above diagram to be small. We will use energy considerations in order to derive the pendulum equation. [Diagram] Let the pendulum rod be attached to a fixed point, O. Let the length of the pendulum be l, and the angle made by the pendulum rod with the vertical axis be x. Then the vertical distance of the bob below the fixed point, O, will be [Equation] . [Equation] Taking the level of the fixed point, O, to be zero gravitational potential energy, the bob, therefore, has [Equation] potential energy at the point where the rod makes the angle, x, with the vertical. The velocity will be measured in the same direction as the angle, x. The arc length is [Equation] [Diagram] so the velocity is [Equation] Hence, the kinetic energy of the bob is [Equation] As the whole system is frictionless, no energy is lost from it, so conservation of energy implies that total energy is conserved. [Equation] Hence, [Equation] where E is a constant. Differentiating both sides with respect to x gives [Equation] Hence, [Equation] Some notes about this process of differentiation. Firstly, energy, E, is a constant, so its derivative is zero. Secondly, we have here differentiated with respect to x, the angular displacement, and not with respect to t, time. Thirdly, you are reminded that the derivative of [Equation goes here - download the original to see it.] Now [Equation] So equation (1) becomes [Equation] Cancelling through the dx terms [Equation] gives [Equation] Hence, [Equation] This is the pendulum equation. [Equation] Since x is small we can approximate x by , hence, on substitution, we obtain [Equation] which is the harmonic approximation to the pendulum equation.
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Contents of
Simple pendulum
1 Simple pendulum
2 Equation of the harmonic oscillator
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