Physics
Rolling Motion
Rolling motion refers to the movement of an object as it rotates and translates simultaneously. This type of motion combines both rotational and translational kinetic energy. When an object rolls, it experiences both linear and angular velocity, allowing it to move forward while also rotating about its axis.
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3 Key excerpts on "Rolling Motion"
eBook - ePub- A. L. Stanford, J. M. Tanner(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
r α relate the translational and rotational variables of points on or within the rolling body. Special properties of the center of mass of a rigid body permit us to treat a body rolling without slipping in a simple fashion:1. We may calculate the net torque ∑τcm about the center of mass of the body, use its moment of inertiaIcmabout an axis through its center of mass, and treat the problem dynamically using ∑τcm =Icmα and ∑F = M a cm , in whichacm= r α.2. We may write the kinetic energy K of the body as K = 1/2Icmω2 + 1/2Mv 2 cm , in whichvcm= r ω.How do we distinguish between rotational problems that are appropriately treated by kinematics or dynamics or energetics? The issue is kinematic if a given problem requires us to relate any of the quantities θ, ω, α, and t among themselves. If the angular acceleration of a system is constant, the equations that relate the rotational kinematic quantities areθ =θ o+ω ot +1 2αt 2ω =ω o+ α tω 2=ω o 2+ 2 α ( θ −θ o)Rotational dynamics problems relate the net torque on a body, its moment of inertia, and its angular acceleration according to ∑τ = Iα . Problems in which we seek torque or angular acceleration (or their corresponding forces or translational acceleration) are usually handled conveniently using rotational dynamics, that is, using ∑τ = Iα . On the other hand, rotational problems that specifically ask for the speed of a body (or its corresponding angular velocity) when the body moves through a vertical distance, exchanging gravitational potential energy and kinetic energy, we should consider using the conservation of energy principle.The correspondence between rotational variables and translational variables permits us to construct appropriate rotational relationships from familiar translational relationships. The following tabulation recalls the correspondence between the variables of translation and rotation:
No longer available |Learn more- Robert A. Pelcovits, Joshua Farkas(Authors)
- 2023(Publication Date)
- Barrons Educational Services(Publisher)
Figure 7.9 . For example, we can easily calculate the speed of a particle at the top of the object to beFigure 7.9The Object’s Kinetic Energy
where Icontact is the rotational inertia about an axis passing through the point of contact with the ground. We can use the parallel axis theorem to compute Icontact :Substituting into the expression for the kinetic energy, we findNoting that the angular velocity of the object about the point of contact is the same as the angular velocity of the object about its center of mass and substituting for vCM = Rω, we obtainThe kinetic energy of combined rotation and translation is simply the sum of the KE of pure rotation and pure translation.Although we do not prove it here, the kinetic energy of a rolling object is always equal to the sum of the kinetic energies of pure rotation and pure translation (even if the motion is not rolling without slipping).The Force Required for Rotation Without Slipping
Recall that if an object rolls without slipping, the following equation is valid.By differentiating, we can make the statement thatcondition necessary but not sufficient for rolling without slippingWhat physical insight is there to be gained from these equations? Consider a few situations.Case 1: If a circular object rolls without slipping at a constant translational speed, the translational acceleration is zero (aCM = 0), and consequently the angular acceleration is zero (α = 0) and the net torque is zero. Thus no net torque is required to keep an object rolling without slipping at a constant speed. (Conservation of angular momentum is enough to keep the object turning at constant speed.) An object that rolls without slipping over a frictionless surface continues to do so, and an object that rolls without slipping at a constant velocity over a surface with friction experiences no frictional force (recall that static friction is variable; you get as much as you need to prevent slipping, up to the maximum value of μs FN ). A free-body diagram of this situation is shown in Figure 7.10
eBook - ePubAP® Physics 1 Crash Course Book + Online
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- Amy Johnson(Author)
- 2016(Publication Date)
- Research & Education Association(Publisher)
Rotational vs. Translational Energya. It is important to know when you will be using translational energy, when you’ll be using rotational energy, and when you would use both.b. For example, a bicycle wheel, sliding along a frictionless table, only has translational kinetic energy, since it is sliding, and not rotating.c. If you had the same bicycle wheel suspended from the ceiling, and the wheel was rotating freely, you would use rotational energy.d. If the bicycle wheel is rolling down the street, it has rotational and translational energy.E. ANGULAR MOMENTUM 1. Angular momentum is the rotational analog of linear momentum. It can be calculated using the equation: L = Ιω, or L = mvR sin θ and is measured in kg · m2 /s.2. Angular momentum is conserved if there is no net external torque, just as linear momentum is conserved if there is no external force.Angular momentum for an object rotating on its own axis can be calculated using L = Ιω. If an object is rotating around an external point, use L = mvR sin θ, where mv is the momentum of the object, R is the distance between the pivot point and the object, and θ is the angle between that R vector and the momentum vector.3. If an ice skater spins with her arms stretched out and then pulls her arms in, you expect that her angular velocity will increase (she will spin faster), but why? Initially, with her arms stretched out, she has a large rotational inertia, and when she pulls her arms in, she has a smaller rotational inertia. There is no net external torque on her while she is pulling her arms in, since the force she is applying to bring her arms in is internal to the system. This means that her angular momentum is conserved, so her initial angular momentum and her final angular momentum are the same:
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