![]() ![]() The moments and product of inertia for an area are plotted as shown and used to construct Mohr’s circle,.Mohr’s Circle for Moments and Products of Inertia I max and I min are the principal moments of inertia of the area about O.The equation for qm defines two angles, 90o apart which correspond to the principal axes of the area about O.At the points A and B, Ix’y’ = 0 and I x’ is a maximum and minimum, respectively.10-9a and 10-9c and adding, it is found that Principal Axes and Principal Moments of Inertia Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. We recommend using aĪuthors: William Moebs, Samuel J. 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 Equation 10.7 relates the angular acceleration to the position and tangential acceleration vectors: We have a vector rotational equivalent of this equation, which can be found by using Equation 10.7 and Figure 10.8. ![]() ![]() The second law Σ F → = m a → Σ F → = m a → tells us the relationship between net force and how to change the translational motion of an object. Deriving Newton’s Second Law for Rotation in Vector FormĪs before, when we found the angular acceleration, we may also find the torque vector. It makes sense that the relationship for how much force it takes to rotate a body would include the moment of inertia, since that is the quantity that tells us how easy or hard it is to change the rotational motion of an object. With this equation, we can solve a whole class of problems involving force and rotation. This is called the equation for rotational dynamics. Thus, if a rigid body is rotating clockwise and experiences a positive torque (counterclockwise), the angular acceleration is positive.Įquation 10.25 is Newton’s second law for rotation and tells us how to relate torque, moment of inertia, and rotational kinematics. Remember the convention that counterclockwise angular acceleration is positive. The term I α I α is a scalar quantity and can be positive or negative (counterclockwise or clockwise) depending upon the sign of the net torque. Substituting this expression into Newton’s second law, we obtain Recall that the magnitude of the tangential acceleration is proportional to the magnitude of the angular acceleration by a = r α a = r α. We apply Newton’s second law to determine the magnitude of the acceleration a = F / m a = F / m in the direction of F → F →. The particle is constrained to move in a circular path with fixed radius and the force is tangent to the circle. Let’s exert a force F → F → on a point mass m that is at a distance r from a pivot point ( Figure 10.37). This raises the question: Is there an analogous equation to Newton’s second law, Σ F → = m a →, Σ F → = m a →, which involves torque and rotational motion? To investigate this, we start with Newton’s second law for a single particle rotating around an axis and executing circular motion. We have thus far found many counterparts to the translational terms used throughout this text, most recently, torque, the rotational analog to force. In this section, we introduce the rotational equivalent to Newton’s second law of motion and apply it to rigid bodies with fixed-axis rotation. We have analyzed motion with kinematics and rotational kinetic energy but have not yet connected these ideas with force and/or torque. ![]() In this section, we put together all the pieces learned so far in this chapter to analyze the dynamics of rotating rigid bodies.
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