Centers of Mass

  1. To the ends of a 6m massless rod are fastened two weights, one with mass 2g and another with mass 7g. Where is the rod’s fulcrum?
  2. Consider a thin straight rod aligned along the \(x\)-axis with left-most endpoint at \(x=1\) and right-most endpoint at \(x=7\).
    1. Suppose the rod has uniform density \(\rho.\) Where is the rod’s fulcrum?
    2. Suppose that at a point \(x\) the rod has point-density \(\rho(x) = 1+x^2.\) Where is the rod’s fulcrum?

    3. Theorem of Pappus

      Consider the region that would be swept out by revolving the rod about the point \(x=0.\) What is the area of this region?
  3. Consider a thin triangular plate, a lamina, embedded in the plane such that, for a specific choice of rectangular coordinate system measured in meters, the corners of the triangle are located at \((1,4)\) and \((7,1)\) and \((7,13).\)
    1. Suppose the lamina has no mass itself, buy to its corners are fastened a weights with masses 2g, 3g, and 4g respectively. Where is the lamina’s center of mass?
    2. What is the area of the region the lamina occupies?
    3. Suppose the lamina has uniform density \(\rho.\)
      1. What is the mass of the lamina?
      2. Where is the lamina’s center of mass? This point, in the case of uniform (constant) density, is called the centroid.
    4. Suppose that at coordinates \((x,y)\) the lamina has density \(\rho(x) = 1+x^2.\) Where is the lamina’s center of mass?
    5. Suppose that at coordinates \((x,y)\) the lamina has point-density \(\rho(x,y) = 1+xy.\) Where is the lamina’s center of mass?

    6. Theorem of Pappus

      Consider the region that would be swept out by revolving the lamina about the line \(x=0.\)
      1. What is the volume of this region?
      2. What is the surface area of this region?
    7. How would all these previous computations have to change if the corner of the lamina located at \((7,13)\) where instead located at \((13,7)?\)