Moment of inertia - Wikipedia
The Idea Behind "Equivalent Mass" true static mass, its rotational inertia, and the ratio. The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid . Mathematically, the moment of inertia of the pendulum is the ratio of the torque due to gravity about .. of gyration, which is the radius of a ring of equal mass around the center of mass of a body that has the same moment of inertia. I will state the equation for torque in a slightly different way than the book equivalent of mass, which is something called the moment of inertia.
Unless acted upon by a net unbalanced force, an object will maintain a constant velocity. Note that "velocity" in this context is defined as a vectorthus Newton's "constant velocity" implies both constant speed and constant direction and also includes the case of zero speed, or no motion.
Since initial publication, Newton's Laws of Motion and by inclusion, this first law have come to form the basis for the branch of physics known as classical mechanics. Kepler defined inertia only in terms of a resistance to movement, once again based on the presumption that rest was a natural state which did not need explanation.
It was not until the later work of Galileo and Newton unified rest and motion in one principle that the term "inertia" could be applied to these concepts as it is today. In fact, Newton originally viewed the phenomenon he described in his First Law of Motion as being caused by "innate forces" inherent in matter, which resisted any acceleration.
Given this perspective, and borrowing from Kepler, Newton attributed the term "inertia" to mean "the innate force possessed by an object which resists changes in motion"; thus, Newton defined "inertia" to mean the cause of the phenomenon, rather than the phenomenon itself. However, Newton's original ideas of "innate resistive force" were ultimately problematic for a variety of reasons, and thus most physicists no longer think in these terms.
As no alternate mechanism has been readily accepted, and it is now generally accepted that there may not be one which we can know, the term "inertia" has come to mean simply the phenomenon itself, rather than any inherent mechanism. Thus, ultimately, "inertia" in modern classical physics has come to be a name for the same phenomenon described by Newton's First Law of Motion, and the two concepts are now considered to be equivalent.
The Effects of Rotational Inertia on Automotive Acceleration
Relativity[ edit ] Albert Einstein 's theory of special relativityas proposed in his paper entitled "On the Electrodynamics of Moving Bodies" was built on the understanding of inertia and inertial reference frames developed by Galileo and Newton.
While this revolutionary theory did significantly change the meaning of many Newtonian concepts such as massenergyand distanceEinstein's concept of inertia remained unchanged from Newton's original meaning in fact, the entire theory was based on Newton's definition of inertia.
However, this resulted in a limitation inherent in special relativity: In an attempt to address this limitation, Einstein proceeded to develop his general theory of relativity "The Foundation of the General Theory of Relativity,"which ultimately provided a unified theory for both inertial and noninertial accelerated reference frames.
As in the comments, this is not an accurate rendering of the equivalence principle as Einstein first pondered it.
Rather, that "gravitation and inertial mass are equal" is how it should read. There are two distinct characteristics of a body - gravitational mass and inertial mass.
The Effects of Rotational Inertia on Automotive Acceleration
The former measure's a body's "coupling strength" to a gravitational field as Newton conceived it - it measures the how much force a "standardized" gravitational field exerts on a body. The latter measures a body's "resistance to shove"; it measures how much impulse you need to impart to a body to change its velocity by a standardized amount. In more experimental terms: The latter is to do with how quickly a body moves after it is shoved by a given standardized impulse shoving machine.
On the face of it, these are very different experiments and two very different properties. And yet, bodies of different inertias fall at the same acceleration in a gravitational field.
If this really is true, then the only way that this can happen is if the two different properties - inertial mass and gravitational mass - are precisely proportional to one another. The first thing to notice is that the torque is a counter-clockwise torque, as it tends to make the rod spin in a counter-clockwise direction.
The rod does not spin because the rope's torque is balanced by a clockwise torque coming from the weight of the rod itself. We'll look at that in more detail later; for now, consider just the torque exerted by the rope. There are three equivalent ways to determine this torque, as shown in the diagram below.
- Inertia and Mass
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Method 1 - In method one, simply measure r from the hinge along the rod to where the force is applied, multiply by the force, and then multiply by the sine of the angle between the rod the line you measure r along and the force.
This is the way the textbook does it; done in this way, the line you measure distance along is called the lever arm. If we give the lever arm the symbol l, from the right-angled triangle it is clear that Using this to calculate the torque gives: Method 3 - In this method, split the force into components, perpendicular to the rod and parallel to the rod.
The component parallel to the rod is along a line passing through the hinge, so it is not trying to make the rod spin clockwise or counter-clockwise; it produces zero torque. The perpendicular component F sinq gives plenty of torque, the size of which is given by: Any force that is along a line which passes through the axis of rotation produces no torque. Note that torque is a vector quantity, and, like angular displacement, angular velocity, and angular acceleration, is in a direction perpendicular to the plane of rotation.
Moment of inertia
The same right-hand rule used for angular velocity, etc. Moment of inertia We've looked at the rotational equivalents of several straight-line motion variables, so let's extend the parallel a little more by discussing the rotational equivalent of mass, which is something called the moment of inertia.
Mass is a measure of how difficult it is to get something to move in a straight line, or to change an object's straight-line motion. The more mass something has, the harder it is to start it moving, or to stop it once it starts.Mass and Inertia
Similarly, the moment of inertia of an object is a measure of how difficult it is to start it spinning, or to alter an object's spinning motion. The moment of inertia depends on the mass of an object, but it also depends on how that mass is distributed relative to the axis of rotation: The moment of inertia of an object depends on where the axis of rotation is. The moment of inertia can be found by breaking up the object into little pieces, multiplying the mass of each little piece by the square of the distance it is from the axis of rotation, and adding all these products up: Fortunately, for common objects rotating about typical axes of rotation, these sums have been worked out, so we don't have to do it ourselves.
A table of some of these moments of inertia can be found on page in the textbook.