Download Rotational Kinematics Energy - General Physics I - Slides | PHYS 130 and more Study notes Physics in PDF only on Docsity! 1 Rotational Kinetic Energy • An object rotating about some axis with an angular speed, ω, has rotational kinetic energy even though it may not have any translational kinetic energy • Each particle has a kinetic energy of • Ki = ½ mivi2 • Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = ωi r Rotational Kinetic Energy, cont • The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles • Where I is called the moment of inertia Rotational Kinetic Energy, final • There is an analogy between the kinetic energies associated with linear motion (K = 1/2 mv 2) and the kinetic energy associated with rotational motion (KR= 1/2 Iω2) • Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object • The units of rotational kinetic energy are Joules (J) Moment of Inertia • The definition of moment of inertia is • The dimensions of moment of inertia are ML2 and its SI units are kg.m2 • We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass Δmi 2 Moment of Inertia, cont • We can rewrite the expression for I in terms of Δm • With the small volume segment assumption, • If ρ is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known Notes on Various Densities • Volumetric Mass Density –> mass per unit volume: • ρ = m / V • Face Mass Density –> mass per unit thickness of a sheet of uniform thickness, t : • σ = ρt • Linear Mass Density –> mass per unit length of a rod of uniform cross-sectional area: • λ = m / L = ρΑ Moment of Inertia of a Uniform Thin Hoop • Since this is a thin hoop, all mass elements are the same distance from the center Moment of Inertia of a Uniform Rigid Rod • The shaded area has a mass • dm = λ dx • Then the moment of inertia is
