So the kinetic energy of the entire rigid body can now be written as, This is the moment of inertia or rotational inertia and is denoted by I. The sum of the products of the masses of the particles by the squares of their distances from the axis is ∑m ir i 2. The total kinetic energy is the sum of the kinetic energies of its particles, and each and every particle in a rigid body moves with the same angular speed Ω (it moves through the same number of degrees of angle per unit time), but the radius r may be different for different particles. It therefore has a linear speed of v = Ωr, and its kinetic energy K is ½ mv 2 = ½ mΩ 2r 2. A particle of mass m at radius r from the axis of rotation moves in a circle of radius r with an angular speed Ω about this axis. If we have a rigid body rotating at an angular speed Ω about a fixed axis, each particle in it will have a certain amount of kinetic energy. Fortunately, the moment of inertia of a uniform solid cylinder about a central diameter is easy to find.
The moment of inertia of the horizontal rod must be subtracted from the total moment in order to be left with the moment of inertia of the wheel alone. From that, we can determine the moment of inertia of the entire apparatus - the wheel and the horizontal rod twisting back and forth. Our vertical rod behaves like a spring, and we aim to discover the torsional spring constant of the rod. The higher the moment of inertia of the wheel, the more slowly the torsional pendulum will twist back and forth.Īnd as long as we keep the deflection to low angles, it does not matter what angle we are actually twisting to one of the characteristics of oscillating harmonic motion is that the period and frequency are independent of amplitude. If we affix a wheel on the end of the vertical rod and twist it to a similarly low angle, it will twist back and forth more slowly. We don’t want to twist so far that we exceed the elastic limit of the rod we want to be well within the range where it springs back and forth in a repeatable manner. If we twist the vertically hanging rod to a relatively small angle without a wheel attached, it will twist back and forth rapidly, and we can measure how many times the horizontal rod welded to it oscillates back and forth in a given amount of time.
Moment of inertia of a circle about its diameter full#
That’s where the torsional pendulum comes in - a device that measures a wheel’s rotational inertia.Įver since I built this torsional pendulum for bicycle wheels and tested the moment of inertia of some wheels in the Jissue of VeloNews, I’ve wanted to provide readers with a full explanation of the physics involved. We can measure the mass of the wheel easily enough, but it is not necessarily the case that the lightest wheel will have the lowest moment of inertia, or vice versa. Rotational inertia, or moment of inertia, is the rotational equivalent of mass this is the quantity that we want to measure to see how much energy it takes to accelerate a wheel. In the case of a wheel, it is probably obvious that it will take more work to accelerate it if the mass is concentrated out at its edge than at its center. This is not true if you drive the object by rotating it then, how the mass is distributed plays an important role in how much energy it takes to move it. If you move a rigid object in a straight line, it does not matter how its mass is distributed the amount of work to move it will be the same. You’ll need them both as Lennard busts out the physics. Grab a cup of coffee and your thinking cap. 18, 2008 issue of VeloNews (and later in the September 2010 issue). Editor’s note: The following is an explanation of the inertia test used in the Aug.
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