Tool · Sandbox

Inertia Explorer

Build a body from standard parts, or load one of the presets, and see its center of mass, inertia tensor, principal axes and inertia ellipsoid update as you change it. Then spin it about any axis and compare \(\wvec\) with \(\Hvec\).

Parts

    No parts yet. Press Add part, or load a preset.

    Select a part in the list to edit it.

    Selected part

    Center of the part (m)

    Direction of the part's own \(z\)-axis

    Mass and center of mass

    Tensor and principal axes about

    Products of inertia are \(I_{xy} = \int xy\,dm\), entered in the tensor with a minus sign. Principal axes are shown in one of their two directions.

    Spin axis

    Arrows show directions only. When \(\Hvec\) is not along \(\wvec\), it turns with the body, and the bearings must supply \(\wvec \times \Hvec\).

    Scene

    Save a picture

    Start over

    The Explorer. Drag the view to rotate it; hold Ctrl (or ⌘) and scroll, or pinch, to zoom. The selected part is drawn in amber and holes in red. Build the body in the Body tab, read its inertia in the Inertia tab, and spin it in the Spin tab.

    How to use the Explorer

    The Body tab holds a list of parts. Each part is a standard body from Lesson 2 (a slender rod, block, thin plate, cylinder, tube, disk, ring, sphere, shell, cone or particle) with a mass, dimensions, a center and a direction. Rods, cylinders, tubes, disks, rings and cones lie along their own \(z\)-axis; blocks and plates can also be turned about it. Tick A hole to subtract a part, as in Lesson 4. Any box accepts an expression such as 0.3/2 or sqrt(2)/2.

    The Inertia tab shows the total mass, the center of mass \(G\), the inertia tensor about the origin \(O\) or about \(G\), and the principal moments and axes about that point (Lessons 4 to 8). The inertia ellipsoid's radius along any axis is proportional to \(1/\sqrt{I}\) about that axis.

    The Spin tab spins the body about an axis through the same point. It shows the moment of inertia about that axis, \(\Hvec = \Imat\wvec\), the angle between \(\Hvec\) and \(\wvec\), and the kinetic energy (Lessons 6 and 7). Your body is saved in this browser, so it is still here next time.

    Ideas to try

    Each idea loads a ready-made body and replaces what is in the Explorer.

    The tennis-racket axis

    A T-handle has three different principal moments about \(G\). Find the intermediate axis, then toss a phone or a book about its three axes to see why that one is unstable (Lesson 8).

    A tilted dumbbell

    Spin a dumbbell whose bar is tilted \(20^\circ\) from the \(xy\)-plane about \(z\). \(\Hvec\) leans away from \(\wvec\) and sweeps a cone (Lesson 6). Change the bar's direction to \((1, 0, 0)\) and the lean disappears.

    Balance a rotor

    A disk sits on a shaft with its axis tilted \(8^\circ\). Its center of mass is on the shaft, yet \(\Hvec\) is not. Set the disk's direction to \((0, 0, 1)\) to balance it dynamically (Lessons 6 and 8).

    A flywheel with holes

    A steel flywheel with a bore and four lightening holes (Worksheet Problem 5). Compare \(I_{zz}\) with and without the holes, and notice that symmetry keeps \(z\) a principal axis (Lessons 4 and 8).

    The bent rod of Example 5.2

    Check the tensor about \(O\) against Example 5.2, then switch to \(G\) and compare with Example 8.3. Spin it about \(z\) at \(10\ \text{rad/s}\) to reproduce Example 6.2.

    A small satellite

    The satellite from the home page. Find its principal axes, then spin it about the axis of maximum moment of inertia: the only spin that stays stable when a real spacecraft dissipates energy (Lesson 8).