Centre of mass of a uniform lamina
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Centre of Mass of a Uniform Lamina
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Our aim is to find the centre of mass of a uniform lamina under the curve y = f(x) between the limits x = a and y = b. The following diagram illustrates this problem. [Diagram goes here - download the original to see it.] We wish to find the centre of mass of the shaded region. A lamina is a thin sheet of material. The sheet is sufficiently thin for its thickness to be ignored, and in the formulae and discussion that follows we treat the lamina as having no thickness whatsoever. Let [Equation goes here - download the original to see it.] be the position of the centre of mass. The centre of mass is defined to be that point in an object where the turning effect of forces is zero. [Diagram goes here - download the original to see it.] Hence, we use the conservation of moments (torque) to find [Equation goes here - download the original to see it.]. However, before we do so let us first state and illustrate the main result of this section. A lamina is bounded by the curve y = f(x), and the lines x = a, x = b, y = 0. Then its centre of mass. [Equation goes here - download the original to see it.] is given by:- [Equation goes here - download the original to see it.] Example (1) A uniform lamina is bounded by the x-axis, the y-axis, the line x = z and the curve y = 4x3.. Find its centre of mass. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The proof of the formula is as follows. To find the centre of mass of a uniform lamina under the curve y = f(x), as shown below. [Diagram goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] be the centre of mass and ? the density of the lamina per unit area. The area of the lamina is [Equation goes here - download the original to see it.] We divide the lamina into n strips of width dx and height y. [Diagram goes here - download the original to see it.] The area of the ith strip is [Equation goes here - download the original to see it.] Since here, mass = density x surface area, the mass of the ith strip is: [Equation goes here - download the original to see it.] The mass of the whole lamina is [Equation goes here - download the original to see it.] By conservation of moments, and by taking moments about the y-axis. Sum of the moments of each strip about its centre of mass = the moment of the whole lamina about [Equation goes here - download the original to see it.] Now [Equation goes here - download the original to see it.] Cancelling through by ?, and taking the limit dx ? 0, we obtain: [Equation goes here - download the original to see it.] To find the position of y-bar, observe that each strip has y-coordinate of centre of mass given by [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Taking moments about the x-axis. [Equation goes here - download the original to see it.]
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Contents of
Centre of mass of a uniform lamina
1 Centre of Mass of a Uniform Lamina
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