11. Mass and Center of Mass

Homework

  1. Four discrete masses are hanging from a balance beam. The masses and positions are indicated below are in kilograms and meters. \[\begin{aligned} m_1&=4 &m_2&=2 &m_3&=5 &m_4&=3 \\ x_1&=-4 &x_2&=-1 &x_3&=3 &x_4&=5 \end{aligned}\]

    The plot shows a balance beam with 4 weights at x equals -4, -1, 3, and 5 with masses 4, 2, 5, and 3, respectively.

    1. Find the total mass.

    2. Find the center of mass.

    3. Find the moment of inertia about \(x_0=2\).

      The plot shows the same balance beam with a triangle added below x = 2 which is the pivot point for computing the moment of inertia.

  2. An \(3\) m bar has linear density given by \(\delta(x)=4x^2\,\dfrac{\text{kg}}{\text{m}}\) where \(x\) is measured from one end.

    The plot shows a bar from x equals 0 to 3, which is shaded from light on the left to dark on the right, showing it is more dense close to x equals 3.

    1. Find the total mass.

    2. Find the center of mass.

    3. Find the moment of inertia about \(x_0=2\).

      The plot shows the same balance beam with a triangle added below x = 2 which is the pivot point for computing the moment of inertia.

  3. A dielectric bar with length \(\dfrac{3\pi}{2}\,\text{cm}\), has linear charge density \(\delta_e(x)=\sin(x)\). Find the net charge on the bar.

    The plot shows a charged bar from x = 0 to x = three pi over two. It is colored blue on the left to show the charge is positive and red on the right to show the charge is negative.

  4. A rod with length \(\sqrt{8}\,\text{m}\) has linear density given by \(\delta(x)=x\sqrt{x^2+1}\;\dfrac{\text{kg}}{\text{m}}\) where \(x\) is measured from one end.

    The plot shows a rod from x equals 0 to square root of 8, which is shaded from light on the left to dark on the right, showing it is more dense when x is close to square root of 8.

    1. Find the total mass.

    2. Find the center of mass.
      Note: This integral is hard. So, set up the integral, but use a computer to evaluate it. (Maple, Mathematics, Wolfram Alpha, etc. Say which program you used.)

    3. Find the moment of inertia about \(x_0=0\).

      The plot shows the same rod with a triangle added below x = 0 which is the pivot point for computing the moment of inertia.

  5. An \(8\) m rod has linear density given by \(\delta(x)=e^{-x}\,\dfrac{\text{kg}}{\text{m}}\) where \(x\) is measured from one end.

    The plot shows a rod from x equals 0 to 8, which is shaded from dark on the left to light on the right, showing it is more dense when x is close to 0.

    1. Find the total mass.

    2. Find the center of mass.

    3. Find the moment of inertia about \(x_0=0\).

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