Classical dynamics of particles & systems (4th ed.). We bring to your attention more detailed tables with formulas for calculating the moment of inertia for the main geometric figures: disk, triangle, solid cylinder, etc. The moment of inertia of a cylinder will be calculated similarly. The moment of inertia for a circle is calculated this way. To simplify the task, a table was created with inertia calculations for simple geometric shapes: circle, square, cylinder, etc. Calculationĭespite its simplicity, the calculation of the moments of inertia for different objects requires knowledge of the integrals, these important tools of higher mathematics.
Using the Steiner formula, we can calculate the moment of inertia of the body relative to any axis of a parallel line that passes through the middle of the figure. For example, you have an object of arbitrary shape, the centrifugal force of which is known. This theorem greatly facilitates the solution of many physical problems associated with inertia. If you write down the above mathematical formula, you get the following: The Huygens-Steiner theorem states: the moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia of a body about an axis passing through the center of mass parallel to an arbitrary axis and the product of the mass of the body by the square of the distance between the axes. Speaking about the moment of inertia, it is impossible not to mention the theorem of two mathematicians Huygens and Steiner, who gave a formulation to the definition of the characteristic of parallel axes. J is the moment of inertia, r is the distance to the axis of rotation.įor a material point of mass m, which rotates around an axis at a distance r, this formula will have the following form: If the body is divided into infinitely small pieces with mass dm, then the moment of inertia will be equal to the sum of the product of these elementary masses by the square of the distance to the axis of rotation.
How to calculate moment of inertia? There is a general equation that helps physicists determine the moment of inertia of any body. It is usually indicated by the letter J and is measured in kilograms multiplied by a square meter. The moment of inertia is a scalar physical quantity, a measure of the inertia of a body when it rotates around an axis. If during the translational motion of some body its mass is a measure of its inertia, then with the rotational movement of the body around its axis, the measure of its inertia will be the quantity that is called the moment of inertia. Area moment of inertia can be used to calculate the stress in a beam due to an applied bending moment at any distance from the neutral axis using the following equation: where is the stress in the beam, y is the distance from the neutral axis passing through the centroid, and I is the area moment of inertia. In the above example, the trolley moves in a straight line and performs a translational motion.
In other words, the more mass the body has, the greater the effect of inertia on it, and the more forces are needed to change the movement of such a heavy body. For example, if in a supermarket two trolleys are pushed hard, one of which will be empty, and the second one loaded with different goods, then later it will be more difficult to stop the trolley loaded with goods due to its larger mass. We are well aware that body mass is a measure of its inertness. Thanks to such everyday examples the concept of inertia is clear, but the term “moment of inertia” requires a more detailed explanation. “It moves by inertia,” we say when we want to emphasize that something is being done without any meaning, but simply because of a habit acquired over the years. Inert persons only do what others tell them, and they do it extremely slowly, without any enthusiasm. For instance, the inert person is the person who does not show any initiative at all. However, the concept of inertia is often used not only in physics but also in our daily lives. As with all calculations care must be taken to keep consistent units throughout.Inertia in physics is the ability of bodies to maintain a state of motion for a certain time in the absence of external forces. The above formulas may be used with both imperial and metric units. Notation and Units Metric and Imperial Units