He exerts a force of 250 N at the edge of the 50.0-kg merry-go-round, which has a 1.50 m radius.Ĭalculate the angular acceleration produced (a) when no one is on the merry-go-round and (b) when an 18.0-kg child sits 1.25 m away from the center. The mass moment of inertia for a general shape.Example 10.7 Calculating the Effect of Mass Distribution on a Merry-Go-RoundĬonsider the father pushing a playground merry-go-round in Figure 10.13. Once we have the dV function in terms of r, we multiply that function by r squared and we will evaluate the integral. The height, radius, and holes in this cylindrical surface may all be changing so this dV term may become quite complex, but technically we could find this for mathematical function for any shape. The rate of change of the volume (dV) will be the cylindrical surface area at a given radius times rate at which that radius is increasing (dr). \įor the polar integral, we need to define dV in terms of a radius (r) moving outwards from the axis of rotation. Approximating a rigid body as an infinite number of infinitely small masses all connected to the axis of rotation, we can sum all the mass times distance squared terms with integration. To relate the moment and angular acceleration in this case, we use integration to add up the infinite number of small mass times distance squared terms. Rather than the massless sticks holding everything in place, the mass is simply held in place by the material around it. Taking the final step, rigid bodies with mass distributed over a volume are like an infinite number of small masses about an axis of rotation. For multiple mass systems, we would simply sum up all the mass times distance squared terms to relate moments and angular accelerations. Taking our situation one step further, if we were to have multiple masses all connected to a central point, the moment and angular acceleration would be related by the sum of all the mass times distance squared terms. This mass times distance squared term (relating the moment and angular acceleration) forms the basis for the mass moment of inertia. A simplified version of this new relationship states that the moment will be equal to the mass times the distance squared times the angular acceleration. If we take these two substitutions and put them into the original F = m a equation, we can wind up with an equation that relates the moment and the angular acceleration for our scenario. We can also relate the linear acceleration of the mass in that the linear acceleration is the angular acceleration times the length of the rod (d). In this case the moment will be related to the force in that the force exerted on the mass times the length of the stick (d) is equal to the moment. To relate the moment and the angular acceleration, we need to start with the traditional form of Newton's Second Law, stating that the force exerted on the point mass by the stick will be equal to the mass times the acceleration of the point mass (F = m a). We are attempting to rotate the mass about it's left end by exerting a moment there. A point mass on the end of a massless stick. We want to relate the moment exerted to the angular acceleration of the stick about this point. Imagine we want to rotate the stick about the left end by applying a moment there. To see why this relates moments and angular accelerations, we start by examining a point mass on the end of a massless stick as shown below. The mass moment of inertia is a moment integral, specifically the second, polar, mass moment integral. The Mass Moment of Inertia and Angular Accelerations This page will only discuss the integration method, as the method of composite parts is discussed on a separate page. Just as with Area Moments of Inertia, the mass moment of inertia can be calculated via moment integrals or via the method of composite parts and the parallel axis theorem. This is represented in an equation with the rotational version of Newton's Second Law. The Mass Moment of Inertia represents a body's resistance to angular accelerations about an axis, just as mass represents a body's resistance to linear accelerations.
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