Skip to main content
Physics LibreTexts

2.6: Three-dimensional Solid Figures. Spheres, Cylinders, Cones.

  • Page ID
    6937
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Sphere, mass \(m\), radius \(a\).


    alt

    The volume of an elemental cylinder of radii \(x\), \( x + \delta x \), height \( 2y \) is \( 4 \pi yx \delta x = 4\pi(a^2-x^2 )^{1/2} x \delta x \). Its mass is \( m \times \frac{4\pi(a^2-x^2)^\frac{1}{2} x \delta x}{\frac{4}{3}\pi a^{3}} = \frac{3m}{a^{3}} \times (a^2-x^2)^\frac{1}{2} x \delta x. \) It' second moment of intertia is \( = \frac{3m}{a^{3}} \times (a^2-x^2)^\frac{1}{2} x^3 \delta x. \) The second moment of inertia of the entire sphere is

    \( = \frac{3m}{a^{3}} \times \int_{0}^{a} (a^2-x^2)^\frac{1}{2} x^3 \delta x = \frac{2}{5} ma^2. \)

    The moment of inertia of a uniform solid hemisphere of mass \( m\) and radius \( a\) about a diameter of its base is also , \( \frac{2}{5} ma^{2} \), because the distribution of mass around the axis is the same as for a complete sphere.

    Exercise \(\PageIndex{1}\)

    A hollow sphere is of mass \( M \), external radius \( a\) and internal radius \( xa \). Its rotational inertia is \( 0.5 Ma^2 \). Show that \(x\) is given by the solution of

    alt

    \( 1 - 5x^3 + 4x^5 = 0 \)

    and calculate \( x\) to four significant figures.

    (Answer = 0.6836.)

    Solid cylinder, mass \( m\), radius \( a\), length \( 2l\)

    alt

    The mass of an elemental disc of thickness \( \delta x \) is \( \frac {m \delta x} {2l} \). Its moment of inertia about its diameter is \( \frac{1}{4} \frac{m \delta x }{2l} a^2 = \frac{m a^2 \delta x }{8l} \). Its moment of inertia about the dashed axis through the centre of the cylinder is \( \frac{m a^2 \delta x }{8l}+ \frac{m \delta x }{2l} x^2 = \frac{m(a^2+4x^2) \delta x}{8l}. \) The moment of inertia of the entire cylinder about the dashed axis is \( 2 \int_{0}^{1} \frac{m(a^2+4x^2) \delta x}{8l} = m(\frac{1}{4}a^2 + \frac{1}{3} l^2)\).

    In a similar manner it can be shown that the moment of inertia of a uniform solid triangular prism of mass \( m \), length \( 2l \), cross section an equilateral triangle of side \(2a \)about an axis through its centre and perpendicular to its length is \(m(\frac{1}{6}a^2 + \frac{1}{3} l^2)\).

    Solid cone, mass \( m\), height \( h\), base radius \( a\).

    alt

    The mass of elemental disc of thickness \( \delta x \) is

    \( m \times \frac{\pi y^2 \delta x}{\frac{1}{3} \pi a^2 h} = \frac{3my^2 \delta x } {a^2h}\).

    Its second moment of inertia about the axis of the cone is

    \( \frac{1}{2} \times \frac{3my^2 \delta x } {a^2h}\times y^2 = \frac{3my^4 \delta x } {2a^2h}\).

    But \( y \) and \( x \) are related through \( y = \frac{ax}{h} \), so the moment of inertia of the elemental disk is

    \( \frac{3ma^2x^4 \delta x } {2h^5}\).

    The moment of inertia of the entire cone is

    \(\frac{3ma^2} {2h^5} \int_{0}^{h}x^{4} dx = \frac{3ma^2} {10}\).

    The following, for a solid cone of mass \(m\), height \(h\), base radius \(a\), are left as an exercise:

    alt


    This page titled 2.6: Three-dimensional Solid Figures. Spheres, Cylinders, Cones. is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.