Moment of inertia of a rod how to#

Point mass going in a circle for what the moment of inertia is, how difficult it's going toīe to angularly accelerate. Of a point mass going in a circle is just the mass times how far that mass is from the axis, squared. Rotating at the same radius like this is, we determined last time that the moment of inertia Has very little mass, you can neglect the mass here. If you've got a heavy ballĬonnected to a string, a very light string that By point mass I just meanĪ mass you could treat as if all the mass were rotating at the same distance from the axis, and that's what's happening here.

Moment of inertia of a rod how to#

Term for angular acceleration, and we figured out how to determine the moment of inertia for a point mass, and you'll hear people say this a lot, 'point mass,' I'm gonna say it a lot. Same role that mass did, it serves as this inertia Larger angular acceleration 'cause you're now dividingīy a smaller number.

You've got a big denominator, you're gonna have a small value, that means this alpha is gonna be small, it's gonna be a smallĪngular acceleration, but if this moment of inertia were small, then it's gonna be easier to rotate, and you'll get a relatively Rotational inertia is big, look it, this is in the denominator. We're dividing by the moment of inertia, we're dividingīy the rotational inertia because that means if this Is gonna be equal to the net torque dividedīy the moment of inertia, or the rotational inertia, In the angular version of Newton's second law, that says that the angular acceleration

Know the moment of inertia is 'cause it'll let you determine how difficult it'll be toĪngularly accelerate something, and remember it shows up So that's what this number is good for, the reason why you wanna To be very difficult to try to get this thing accelerating, but if the moment of inertia is small, it should be very easy, relatively easy to get this thing angularly accelerating. System has a large moment of inertia, it's going In other words, how much something's going to resist being angularly accelerated, so being sped up in its This moment of inertia is really just the rotational inertia. 'cause this is something that people get confused about a lot. Integrating from -L/2 to +L/2 from the center includes the entire rod.Talk some more about the moment of inertia, Since the totallength L has mass M, then M/L is the proportion of mass to length and the masselement can be expressed as shown. To perform the integral, it is necessary to express eveything in the integral in terms of one variable, in this case the length variable r. The moment of inertia calculation for a uniform rod involves expressing any mass element in terms of a distanceelement dr along the rod. When the mass element dm is expressed in terms of a length element dr along the rod and the sum taken over the entire length, the integral takes the form: The general form for the moment of inertia is: The resulting infinite sum is called an integral. The moment of inertia of a point mass is given by I = mr 2, but the rod would have to be considered to be an infinite number of point masses, and each must be multiplied by the square of its distance from the axis. If the thickness is not negligible, then the expression for I of a cylinder about its end can be used.Ĭalculating the moment of inertia of a rod about its center of mass is a good example of the need for calculus to deal with the properties of continuous mass distributions. The moment of inertia about the end of the rod is The moment of inertia about the end of the rod can be calculated directly or obtained from the center of mass expression by use of the Parallel axis theorem. HyperPhysics***** Mechanics ***** Rotationįor a uniform rod with negligible thickness, the moment of inertia about its center of mass is This process leads to the expression for the moment of inertia of a point mass. This provides a setting for comparing linear and rotational quantities for the same system. If the mass is released from a horizontal orientation, it can be described either in terms of force and accleration with Newton's second law for linear motion, or as a pure rotation about the axis with Newton's second law for rotation. Moment of Inertia Rotational and Linear ExampleĪ mass m is placed on a rod of length r and negligible mass, and constrained to rotate about a fixed axis.