Often though, one may use the term "moment of inertia of circle", missing to specify an axis. Where A is the area of the shape and y the distance of any point inside area A from a given axis of rotation.įrom this definition it becomes clear that the moment of inertia is not a property of the shape alone but is always related to an axis of rotation. The second moment of area of any planar, closed shape is given by the following integral: Typical units for the moment of inertia, in metric, are: Typical units for the moment of inertia, in the imperial system of measurements are: By definition, the moment of inertia is the second moment of area, in other words the integral sum of cross-sectional area times the square distance from the axis of rotation, hence its dimensions are ^4. In fact, this is true for the moment of inertia of any shape, not just the circle. Since those are lengths, one can expect that the units of moment of inertia should be of the type: ^4. The above equations for the moment of inertia of circle, reveal that the latter is analogous to the fourth power of circle radius or diameter. Summer Olympics, here he comes! Confirmation of these numbers is left as an exercise for the reader.The moment of inertia of circle with respect to any axis passing through its centre, is given by the following expression:Įxpressed in terms of the circle diameter D, the above equation is equivalent to: The father would end up running at about 50 km/h in the first case. In terms of revolutions per second, these angular velocities are 2.12 rev/s and 1.41 rev/s, respectively.
If, for example, the father kept pushing perpendicularly for 2.00 s, he would give the merry-go-round an angular velocity of 13.3 rad/s when it is empty but only 8.89 rad/s when the child is on it. The angular accelerations found are quite large, partly due to the fact that friction was considered to be negligible. The angular acceleration is less when the child is on the merry-go-round than when the merry-go-round is empty, as expected. Because the force is perpendicular to r, an accelerationa=\frac. To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point, as shown in Figure 2.
If you push on a spoke closer to the axle, the angular acceleration will be smaller. The more massive the wheel, the smaller the angular acceleration. The greater the force, the greater the angular acceleration produced. Force is required to spin the bike wheel. There are, in fact, precise rotational analogs to both force and mass.įigure 1. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration another implication is that angular acceleration is inversely proportional to mass. Furthermore, we know that the more massive the door, the more slowly it opens. For example, we know that a door opens slowly if we push too close to its hinges. In fact, your intuition is reliable in predicting many of the factors that are involved. If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in Figure 1.