Dynamics of Rotational Motion: Rotational Inertia Quiz

PDF page 448: the force is perpendicular to , an acceleration is obtained in the direction of . We can rearrange this equation such that and then look for ways to relate this expression to expressions for rotational quantities. We note that , and we substitute this expression into , yielding 10.40 Recall that torque is the turning effectiveness of a force. In this case, because is perpendicular to , torque is simply . So, if we multiply both sides of the equation above by , we get torque on the left-hand side. That is, 10.41 or 10.42 This last equation is the rotational analog of Newton’s second law ( ), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and is analogous to mass (or inertia). The quantity is called the rotational inertia or moment of inertia of a point mass a distance from the center of rotation.

PDF page 448: the force is perpendicular to , an acceleration is obtained in the direction of . We can rearrange this equation such that and then look for ways to relate this expression to expressions for rotational quantities. We note that , and we substitute this expression into , yielding 10.40 Recall that torque is the turning effectiveness of a force. In this case, because is perpendicular to , torque is simply . So, if we multiply both sides of the equation above by , we get torque on the left-hand side. That is, 10.41 or 10.42 This last equation is the rotational analog of Newton’s second law ( ), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and is analogous to mass (or inertia). The quantity is called the rotational inertia or moment of inertia of a point mass a distance from the center of rotation.

PDF page 448: Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.

PDF page 447: angular acceleration. If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in Figure 10.9. In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely

PDF page 448: proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.

PDF page 447: angular acceleration. If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in Figure 10.9. In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely

PDF page 449: Before we can consider the rotation of anything other than a point mass like the one in Figure 10.10, we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia of an object to be the sum of for all the point masses of which it is composed. That is, . Here is analogous to in translational motion. Because of the distance , the moment of inertia for any object depends on the chosen axis. Actually, calculating is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore

PDF page 449: Before we can consider the rotation of anything other than a point mass like the one in Figure 10.10, we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia of an object to be the sum of for all the point masses of which it is composed. That is, . Here is analogous to in translational motion. Because of the distance , the moment of inertia for any object depends on the chosen axis. Actually, calculating is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore

PDF page 449: , where is its total mass and its radius. (We use and for an entire object to

PDF page 449: integration over the continuous body. Note that has units of mass multiplied by distance squared ( ), as we might expect from its definition. The general relationship among torque, moment of inertia, and angular acceleration is 10.43 or

PDF page 449: where net is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation. Such torques are either positive or negative and add like ordinary numbers. The relationship in is the rotational analog to Newton’s second law and is very generally applicable. This equation is actually valid for any torque, applied to any object, relative to any axis. As we might expect, the larger the torque is, the larger the angular acceleration is. For example, the harder a child pushes on a merry-go-round, the faster it accelerates. Furthermore, the more massive a merry-go-round, the slower it accelerates for the same torque. The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. But there is an additional twist. 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. 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 mass is the same in both cases, but the moment of inertia is much larger when the children are at the edge.

PDF page 450: Consider the father pushing a playground merry-go-round in Figure 10.12. 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. Calculate 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. Consider the merrygo-round itself to be a uniform disk with negligible retarding friction.

PDF page 450: Consider the father pushing a playground merry-go-round in Figure 10.12. 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. Calculate 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. Consider the merrygo-round itself to be a uniform disk with negligible retarding friction.

PDF page 451: Solution for (b) We expect the angular acceleration for the system to be less in this part, because the moment of inertia is greater when the child is on the merry-go-round. To find the total moment of inertia , we first find the child’s moment of inertia by considering the child to be equivalent to a point mass at a distance of 1.25 m from the axis. Then, 10.50 The total moment of inertia is the sum of moments of inertia of the merry-go-round and the child (about the same axis). To justify this sum to yourself, examine the definition of : 10.51 Substituting known values into the equation for gives

PDF page 451: Discussion The angular acceleration is less when the child is on the merry-go-round than when the merry-go-round is empty, as expected. The angular accelerations found are quite large, partly due to the fact that friction was considered to be

PDF page 452: Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple? Solution No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.