That is,. The quantity mr 2 is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of rotation. Figure 2. An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force.
A force F is applied to the object perpendicular to the radius r , causing it to accelerate about the pivot point. The force is kept perpendicular to r. Before we can consider the rotation of anything other than a point mass like the one in Figure 2, 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 I of an object to be the sum of mr 2 for all the point masses of which it is composed.
Here I is analogous to m in translational motion. Because of the distance r , the moment of inertia for any object depends on the chosen axis. Actually, calculating I 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. We use M and R for an entire object to distinguish them from m and r for point masses. In all other cases, we must consult Figure 3 note that the table is piece of artwork that has shapes as well as formulae for formulas for I that have been derived from integration over the continuous body.
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. 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. In statics, the net torque is zero, and there is no angular acceleration. Consider the father pushing a playground merry-go-round in Figure 4.
He exerts a force of N at the edge of the Calculate the angular acceleration produced a when no one is on the merry-go-round and b when an Consider the merry-go-round itself to be a uniform disk with negligible retarding friction. Figure 4. A father pushes a playground merry-go-round at its edge and perpendicular to its radius to achieve maximum torque. To find the torque, we note that the applied force is perpendicular to the radius and friction is negligible, so that.
It is also called angular motion or circular motion. The motion may be uniform i. An object can be rotating while also experiencing linear motion; just consider a football spinning like a top as it also arcs through the air, or a wheel rolling down the street. Scientists consider these kinds of motion separately because separate equations but again, tightly analogous are required to interpret and explain them. It's actually useful to have a special set of measurements and calculations to describe rotational motion of those objects as opposed to their translational or linear motion, because you often get a brief refresher in things like geometry and trigonometry, subjects it is always good for the science-minded to have a firm handle on.
While the ultimate non-acknowledgment of rotational motion might be "Flat Earthism," it is actually pretty easy to miss even when you're looking, perhaps because many people's minds are trained to equate "circular motion" with "circle. Such motion is all around us, with examples including rolling balls and wheels, merry-go-rounds, spinning planets and elegantly twirling ice-skaters.
Examples of motions that may not seem like rotational motion, but in fact are, include see-saws, opening doors and the turn of a wrench.
As noted above, because in these cases the angles of rotation that are involved are often small, it's easy to not filter this in your mind as angular motion. Think for a moment about the motion of a cyclist with respect to the "fixed" ground.
Last reviewed: September The motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity; also, the rotation of a particle about a fixed point in space. Rotational motion is illustrated by 1 the fixed speed of rotation of the Earth about its axis Fig. This article's discussion of rotational motion is limited to the circular motion exhibited by the first and second examples. When two torques of equal magnitude act in opposing directions, there is no net torque and no angular acceleration, as you can see in the following video.
If zero net torque acts on a system spinning at a constant angular velocity, the system will continue to spin at the same angular velocity. This video defines torque in terms of moment arm which is the same as lever arm.
It also covers a problem with forces acting in opposing directions about a pivot point. If the net torque acting on the ruler from the example was positive instead of zero, what would this say about the angular acceleration? What would happen to the ruler over time? A deep-sea fisherman uses a fishing rod with a reel of radius 4. A big fish takes the bait and swims away from the boat, pulling the fishing line from his fishing reel. How long does it take the reel to come to a stop? We are asked to find the time t for the reel to come to a stop.
We solve the equation algebraically for t , and then insert the known values. The time to stop the reel is fairly small because the acceleration is fairly large. Fishing lines sometimes snap because of the forces involved, and fishermen often let the fish swim for a while before applying brakes on the reel.
A tired fish will be slower, requiring a smaller acceleration and therefore a smaller force. Consider the man pushing the playground merry-go-round in Figure 6. He exerts a force of N at the edge of the merry-go-round and perpendicular to the radius, which is 1. How much torque does he produce? Assume that friction acting on the merry-go-round is negligible.
To find the torque, note that the applied force is perpendicular to the radius and that friction is negligible. The man also maximizes his torque by pushing at the outer edge of the merry-go-round, so that he gets the largest-possible lever arm. What is its angular acceleration? Use the Check Your Understanding questions to assess whether students master the learning objectives of this section.
If students are struggling with a specific objective, these questions will help identify which objective is causing the problem and direct students to the relevant content.
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