![]() ![]() If the driver depresses the accelerator slowly, causing the car to move forward, then the tires roll without slipping. If the driver depresses the accelerator to the floor, such that the tires spin without the car moving forward, there must be kinetic friction between the wheels and the surface of the road. For example, we can look at the interaction of a car’s tires and the surface of the road. People have observed rolling motion without slipping ever since the invention of the wheel. You may also find it useful in other calculations involving rotation. Understanding the forces and torques involved in rolling motion is a crucial factor in many different types of situations.įor analyzing rolling motion in this chapter, refer to Figure 10.20 in Fixed-Axis Rotation to find moments of inertia of some common geometrical objects. Think about the different situations of wheels moving on a car along a highway, or wheels on a plane landing on a runway, or wheels on a robotic explorer on another planet. Rolling motion is that common combination of rotational and translational motion that we see everywhere, every day. Use energy conservation to analyze rolling motion.Calculate the static friction force associated with rolling motion without slipping.Find the linear and angular accelerations in rolling motion with and without slipping.Explain how linear variables are related to angular variables for the case of rolling motion without slipping.Describe the physics of rolling motion without slipping.Lots of examples.By the end of this section, you will be able to: The best way to learn how to do this is by example. What can I say about the perpendicular axis theorem other than it's interesting. What if an object isn't being rotated about the axis used to calculate the moment of inertia? Apply the parallel axis theorem. Where α is a simple rational number like 1 for a hoop, ½ for a cylinder, or ⅖ for a sphere. When you are done with all of this, you oftentimes end up with a nice little formula that looks something like this… These methods can be used to find the moment of inertia of things like spheres, hollow spheres, thin spherical shells and other more exotic shapes like cones, buckets, and eggs - basically, anything that might roll and that has a fairly simple mathematical description. Or this for stacked disks and washers I = Something like for nested, cylindrical shells… I = When shapes get more complicated, but are still somewhat simple geometrically, break them up into pieces that resemble shapes that have already been worked on and add up these known moments of inertia to get the total.įor slightly more complicated round shapes, you may have to revert to an integral that I'm not sure how to write. ![]() This method can be applied to disks, pipes, tubes, cylinders, pencils, paper rolls and maybe even tree branches, vases, and actual leeks (if they have a simple mathematical description). The volume of each infinitesimal layer is then…įor many cylindrical objects, you basically start with something like this… I = Imagine a leek.Įach layer of the leek has a circumference 2π r, thickness dr, and height h. The other easy volume element to work with is the infinitesimal tube. Note that although the strict mathematical description requires a triple integral, for many simple shapes the actual number of integrals worked out through brute force analysis may be less. This is the way to find the moment of inertia for cubes, boxes, plates, tiles, rods and other rectangular stuff. When an object is essentially rectangular, you get a set up something like this… I = ![]() The volume of each infinitesimal piece is… The pieces are dx wide, dy high, and dz deep. The infinitesimal box is probably the easiest conceptually. In practice, this may take one of two forms (but it is not limited to these two forms). The infinitesimal quantity dV is a teeny tiny piece of the whole body. In practice, for objects with uniform density ( ρ = m/ V) you do something like this… I =įor objects with nonuniform density, replace density with a density function, ρ( r). You add up (integrate) all the moments of inertia contributed by the teeny, tiny masses ( dm) located at whatever distance ( r) from the axis they happen to lie. It works like mass in this respect as long as you're adding moments that are measured about the same axis.įor an extended body, replace the summation with an integral and the mass with an infinitesimal mass. Say it, kilogram meter squared and don't say it some other way by accident.įor a collection of objects, just add the moments. It's a scalar quantity (like its translational cousin, mass), but has unusual looking units. Logic behind the moment of inertia: Why do we need this? ![]()
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