Therefore, the total (translational plus rotational) Kinetic Energy for the rolling ring is KE = m There would be the rotational Kinetic Energy, and since point O is moving relative to the horizontal surface, there would have to be translational Kinetic Energy as well: Let’s analyze the motion from the center of mass position, Point O. It has rotational speed, ω, around Point O, horizontal center of mass speed, v, and radius, r. How would you determine the Kinetic Energy for an object that is rotating and translating? Let’s look at the ring moving from Position A to Position C as it rolls along the surface as shown at right. The axis through Point A is going through the center of mass. Let’s use the long, thin rod as shown below to calculate I about the axis through Point B. The Moment of Inertia of an object of mass, M, about an axis a distance, d, from the center of mass and parallel to an axis going through the center of mass is calculated by the following: To determine this Moment of Inertia we use the Parallel Axis Theorem that states: An example would be twirling a long, thin rod around an axis that is ¼ of the way from the center of mass. The total resistance to rotation around the vertical center line has to include the rod and the 2 point masses to get the total Moment of Inertia for the system.Ĭonsider an object that is rotating around an axis that does not go through the center of mass of the object. For example, let’s look at a baton that is twirled in the middle consisting of a thin rod (mass M), and a point mass, m, on each end as shown below. Using the definition for torque: τ = r x F , which is just the left side then,Ĭonsider an obect object that is made up of more fundamental objects as depicted in the previous table. Substituting from Rotational Kinematics: a tan = r The equation using Newton’s 2 nd Law in the tangential direction would be F = m Since the motion we care about is in the horizontal plane we can ignore the gravitational and normal force on the mass. Since the centripetal force does not affect tangential force, we are not including it in this discussion. In this derivation, our focus is on the effect of the tangential force. * We know that a circular motion requires an inward or centripetal force. A force, F, tangent to a circle of radius, r, is applied to the mass without frictional effects as shown at right. The object's rotation speed may be increasing, decreasing, or remaining constant.Ĭonsider an object of mass, m, on a horizontal surface connected by a massless rod to a center point O. There are 8 ready-to-use problem sets on the topic of Rotational Dynamics. These problem sets focus on the analysis of situations involving a rigid object or objects rotating in either a clockwise or counterclockwise direction about a given point. Rotational Dynamics: Problem Set Overview
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