In the cartoon we have a wagonwheel ready to roll without slipping from a height “h.” The radius of the wagonwheel is “r” and it is made of a thin hoop of uniform mass density. When it is released from rest, it will roll without slipping toward the loop-the-loop. The loop-the-loop has a radius “R.” The wagonwheel sidesteps the loop-the-loop at first with a negligible offset in the z-direction in order to enter it. You may assume that the wagonwheel radius “r” is much much less than “R” so that the point of contact of the wagonwheel can be treated as if it is in the center-of-mass location of the wheel for brevity. a) What is the moment of inertia of the wagonwheel? Express your answer algebraically here. b) What is the velocity of the wagonwheel at the top of the loop if it is locked to the path and therefore not allowed to fall off? Express your answer algebraically here. c) Suppose the height “h” of release is 25 meters and the radius “R” of the loop-the-loop is 10 meters. If the wagonwheel is unlocked and is now allowed to fall off, will it fall off the loop-the-loop and crash before reaching the top? Notice no specific value for the radius “r” of the wagonwheel. It is not needed, nor is the wagonwheel’s mass. (Special HINT: You may want to draw a force diagram at the top of the loop and sum the forces on the wagonwhe
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
In the cartoon we have a wagonwheel ready to roll without slipping from a height “h.” The radius of the wagonwheel is “r” and it is made of a thin hoop of uniform mass density. When it is released from rest, it will roll without slipping toward the loop-the-loop. The loop-the-loop has a radius “R.” The wagonwheel sidesteps the loop-the-loop at first with a negligible offset in the z-direction in order to enter it. You may assume that the wagonwheel radius “r” is much much less than “R” so that the point of contact of the wagonwheel can be treated as if it is in the center-of-mass location of the wheel for brevity.
a) What is the moment of inertia of the wagonwheel? Express your answer algebraically here.
b) What is the velocity of the wagonwheel at the top of the loop if it is locked to the path and therefore not allowed to fall off? Express your answer algebraically here.
c) Suppose the height “h” of release is 25 meters and the radius “R” of the loop-the-loop is 10 meters. If the wagonwheel is unlocked and is now allowed to fall off, will it fall off the loop-the-loop and crash before reaching the top? Notice no specific value for the radius “r” of the wagonwheel. It is not needed, nor is the wagonwheel’s mass. (Special HINT: You may want to draw a force diagram at the top of the loop and sum the forces on the wagonwheel.)
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