Simple pendulum
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Equation of the harmonic oscillator
This is the equation of a harmonic oscillator, as rearrangement will quickly show [Equation] That is, acceleration is proportional to displacement and acts in the opposite direction. The harmonic approximation to the pendulum equation, therefore, has solution [Equation] In this equation x represents the angular displacement. But the same equation could equally describe the motion of the pendulum if we used the horizontal displacement of the bob from the vertical axis. [Diagram] This is because, when the angular displacement, here q, is small, then the horizontal displacement of the bob from the vertical axis is also approximately given by [Equation] which means that the same equation Equation] escribes the motion of a pendulum whether x measures the angular displacement or the horizontal displacement, provided the displacements are small. xampl Diagram] bob is suspended from a fixed point A by means of a light inextensible string, which has length 5m. The bob is pulled up so that it makes an angle a with the downward vertical line through A. It is then released from rest. Let q be the angle made by the string with the downward vertical through A after t seconds. (i) Find the transverse component of the acceleration of the particle at time t. (ii) Obtain a differential equation involving Equation]./iii) Use the approximation [Equation goes here - download the original to see it.]to show that when a is small, and when air-resistance is ignored, the bob performs simple harmonic motion. (iv) Find the approximate period of the oscillation, giving your answer to 2 significant figures. Solution [Diagram] (i) [Equation goes here - download the original to see it.]. (ii)Let x denote the displacement of the bob along the arc of its path from the vertical axis. Then [Equation] The acceleration of the bob in the transverse direction is in the opposite direction to the displacement, and is given by [Equation] [Equation] The negative sign indicates that the force acts in the opposite direction to the displacement. On simplifying, we obtain [Equation] [Equation] which is a differential equation involving [Equation](iii) [Equation] (iv)The solution of this equation is [Equation]
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Contents of
Simple pendulum
1 Simple pendulum
2 Equation of the harmonic oscillator
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