Rotational motion11/30/2023 ![]() For a whole object, there may be many torques. The left hand side of the equation is torque. Note that the radial component of the force goes through the axis of rotation, and so has no contribution to torque. If we multiply both sides by r (the moment arm), the equation becomes However, we know that angular acceleration, \(\alpha\), and the tangential acceleration atan are related by: If the components for vectors \(A\) and \(B\) are known, then we can express the components of their cross product, \(C = A \times B\) in the following wayįurther, if you are familiar with determinants, \(A \times B\), is ![]() \(A \times B = A B \sin(\theta)\) Figure CP2: \(B \times A = D\) If we let the angle between \(A\) and \(B\) be, then the cross product of \(A\) and \(B\) can be expressed as Then, their cross product, \(A \times B\), gives a third vector, say \(C\), whose tail is also at the same point as those of \(A\) and \(B.\) The vector \(C\) points in a direction perpendicular (or normal) to both \(A\) and \(B.\) The direction of \(C\) depends on the Right Hand Rule. That is, for the cross of two vectors, \(A\) and \(B\), we place \(A\) and \(B\) so that their tails are at a common point. The cross product of two vectors produces a third vector which is perpendicular to the plane in which the first two lie. The cross product, also called the vector product, is an operation on two vectors. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. We recommend using aĪuthors: Paul Peter Urone, Roger Hinrichs ![]() Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the The mass is the same in both cases, but the moment of inertia is much larger when the children are at the edge. For example, it will be much easier to accelerate a merry-go-round full of children if they stand close to its axis than if they all stand at the outer edge. The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates. The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. ![]() ![]() Furthermore, the more massive a merry-go-round, the slower it accelerates for the same torque. For example, the harder a child pushes on a merry-go-round, the faster it accelerates. This equation is actually valid for any torque, applied to any object, relative to any axis.Īs we might expect, the larger the torque is, the larger the angular acceleration is. To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F F size 12 is the rotational analog to Newton’s second law and is very generally applicable. 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. Figure 10.10 Force is required to spin the bike wheel. ![]()
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