Centre of mass of a uniform solid of revolution
DOWNLOAD
FREE
|
Centre of Mass of a Uniform Solid of Revolution
Our aim is to find the centre of mass of a uniform solid of revolution under the curve y = f(x) between the limits x = a and x = b. The following diagram illustrates this problem. [Diagram goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] be the centre of mass. By symmetry - since the solid is formed by revolution about the x-axis, [Equation goes here - download the original to see it.] The result for is: [Equation goes here - download the original to see it.] Before we prove this result we illustrate its application. Example (1) Prove that the centre of mass of a solid hemisphere of radius a is 3a/8 from the plane face. [Diagram goes here - download the original to see it.] The x-coordinate of the centre of mass is [Equation goes here - download the original to see it.] We will now prove the result. [Diagram goes here - download the original to see it.] To find the centre of mass of a uniform solid of revolution under the curve y = f(x) between the limits x = a and y = b. Let [Equation goes here - download the original to see it.] be the centre of mass and ? the density of the solid per unit volume. The volume of the solid of revolution is: [Equation goes here - download the original to see it.] We divide the revolution into n strips of width . The height of each strip corresponding to ordinate xi is yi. [Diagram goes here - download the original to see it.] The volume of each strip is [Equation goes here - download the original to see it.] Since mass = density x volume. The mass of the ith strip is [Equation goes here - download the original to see it.] The mass of the whole solid is [Equation goes here - download the original to see it.] Taking moments about the y-axis. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Therefore, [Equation goes here - download the original to see it.] We conclude with a further example of applications of this result. Example (2) Prove that the centre of mass of a solid cone of height h is h/4 from its base. [Diagram goes here - download the original to see it.] A solid cone is formed by revolving the straight line y = f(x) as shown. Let the base have area a. To find the equation of y = f(x) note that it is a straight line with equation y = mx + c. The gradient is -a/h and the intercept is a. Hence it is: [Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it.] And [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
|
Contents of
Centre of mass of a uniform solid of revolution
1 Centre of Mass of a Uniform Solid of Revolution
|
