Compound pendulum
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Compound pendulum
A compound pendulum is a body made of two or more masses connected together and swinging freely about a smooth horizontal axis. The simplest case would be when just two masses are connected. [Diagram goes here - download the original to see it.] However, there may be more than two masses, and a compound pendulum could be made of an irregular lamina. [Diagram goes here - download the original to see it.] This can be considered to be a compound pendulum, since the lamina can be thought of as being made up of a large number of smaller segments, each with their own mass. The problem of the compound pendulum is essentially to find its period of oscillation (or equivalently, to find its angular frequency). This problem is solved by reducing the compound pendulum to a simple pendulum to which it is equivalent. This is done through energy considerations as shown in the example that follows. The crucial things to remember are [Equation] [Equation] he point of this type of problem is captured by the observation that the expression for the moment of inertia of an object, I, replaces mass in the usual formula for linear kinetic energy. Since the moment of inertia has replaced mass, we need to be able to find it. In problems of this type we usually start with the standard moments of inertia for objects such as discs, rods and rectangular laminas, and derive the moment of inertia for the composite body using the parallel axis theorem. Hence, we need to keep the parallel axis theorem also in our minds: 3. parallel axis theorem states that, supposing the moment of inertia of a body M about an axis passing through its centre of mass is Mk2, then its moment of inertia about an axis parallel to this first axis but at a distance d from it is [Equation]. [Equation] [Equation] This assumes that the aim of the problem is to derive the equation of simple harmonic motion from the data presented in the question. However, other questions involving energy conversions can be set, but these follow the same principle that total energy is conserved. Just a note about the use of the symbol . This symbol is used here to denote angular frequency. This is the angle swept out per unit time, and is a constant for the particular oscillation described by the simple or compound pendulum (or object under simple harmonic motion generally). This symbol is also used to denote angular velocity [Equation] which is a variable that depends on time, and is maximum as the pendulum passes through the vertical and zero at the points of maximum angular amplitude. Because of the possible confusion of having the same symbol to designate two different mathematical concepts, we advise you in this context to use [Equation goes here - download the original to see it.] to denote the angular velocity, and to denote the angular frequency. Example (i) A disc of mass m and radius r is suspended from a point A on its rim. The centre of the disc is the point O. Initially, the disc is hanging so that the line OA is vertical. Assuming that the point of contact is frictionless, and that the disc can be treated as a uniform lamina, show that when the line OA is displaced slightly from the vertical by an angle a that the disc oscillates with simple harmonic motion, and find an expression for its angular frequency. (ii) The disc is now stopped and positioned so that the line OA makes an angle of [Equation] with the vertical. It is then released. Find the angular velocity of the disc as OA first becomes vertical. [Diagram goes here - download the original to see it.] Solution To solve this problem we must first find an expression for the moment of inertia of the disc about the pivot point A. The parallel axis theorem states that, supposing the moment of inertia of a body M about an axis passing through its centre of mass is Mk2, then its moment of inertia about an axis parallel to this first axis but at a distance d from it is [Equation]. The moment of inertia of the disc is [Equation] The disc is at a distance r from the pivot point, A. Hence [Equation] [Equation], then the kinetic energy of the disc when it makes an angle with the vertical is given by [Equation] [Diagram goes here - download the original to see it.] At this point the centre of the disc, which is its centre of gravity, has risen [Equation], so the gravitational potential energy of the disc is [Equation] Since the system is frictionless, total energy is conserved, so [Equation] On differentiating both sides [Equation] (The derivative of E¸ the constant energy, is zero.) On dividing by [Equation] and rearranging we obtain [Equation] This is the equation of motion for the body. As already indicated, here [Equation] Hence, [Equation] If the angle of displacement is small, then [Equation] hence [Equation] This is the equation of simple harmonic motion, and has angular frequency [Equation] 2. [Diagram goes here - download the original to see it.] [Equation] [Equation] Example (2) A uniform rod, AB, has mass 6m and length 4l. It is suspended so that it can move freely about a smooth horizontal axis through A. A further particle of mass 4m is attached to it at B. From a position of hanging at rest in equilibrium, it is given an initial angular velocity of [Equation]. Find the maximum height to which the line AB rises. Solution The moment of inertia of a rod about an axis perpendicular to the rod and 2l distant from its centre of mass is given by [Equation] Here the rod has a mass [Equation] and a length, [Equation], hence its moment of inertia is [Equation] The moment of inertia of a particle at a distance r from an axis of rotation is given by [Equation] [Equation] The total moment of inertia for the rod and particle is, therefore [Equation] [Diagram goes here - download the original to see it.] Let the angle made by the rod AB with the vertical at the moment of instantaneous rest be q. Then, when the rod has risen to this height, it has acquired [Equation goes here - download the original to see it.] joules of gravitational potential energy. At the same time the particle has acquired [Equation goes here - download the original to see it.] joules of gravitational potential energy. The total system has [Equation] joules of gravitational potential energy. The initial kinetic energy is [Equation] The kinetic energy is converted to gravitation potential energy, hence [Equation] On substitution [Equation]
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Contents of
Compound pendulum
1 Compound pendulum
2 Compound pendulum and its equivalent simple pendulum
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