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You can see the r 2-ness of this quantity already. The 3x3 matrix m(r 2? - rr T) is the moment of inertia, and usually called I. Where ? is the 3x3 identity matrix and r T is the matrix transpose of the vector r, so that rr T is a 3x3 matrix. whose motion is purely due to a rotation about some axis), v = ? × r, where ? is angular velocity, represented as a pseudo-vector.
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Where r is the particle's position and p is its momentum. (It's salient for the same sort of reasons that momentum is salient: it's conserved for force-free systems, and it's temporal derivative yields force-like quantities - in the rotational case, torque.) For a single particle, this quantity is given by You can make similar arguments about the angular momentum L = r x p = I ωĪ salient physical quantity for rotating systems is the angular momentum. There are more advanced concepts like the moment of inertia tensor which gives you the moment of inertia along an arbitrary axis of rotation, but this is the basic concept. I = Σ mi ri² (in the discrete case where the body consists of a discrete number of masses mi at radii ri) or ∫ r² dm in the continuous case. Its rotational kinetic energy will then be 1/2 I ω² again. Now if you have an extended body rotating at angular velocity ω (and you don't care how fast the point masses it consists of are moving individually - that varies depending how far they are from the rotational axis), you can calculate its total moment of inertia I by summing all the contributions mr² of all points masses it consists of. Now you define the moment of inertia of the point particle to be I = mr², so T = 1/2 I ω². You can express the velocity as v = rω and the kinetic energy can then be written as T = 1/2 m r² ω². The angular velocity is ω = v/r where r is the radius of that circle. Sitemap Page was generated in 0.What's the kinetic energy of a point particle going around (uniformly) in a circle? T = 1/2 m v² where v is the tangential velocity.
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What is the moment of inertia of a circle? What is the moment of inertia of a circle?