In the cartoon we have a hoop ready to roll without slipping from a height “h.” The radius of the hoop is “r” and its mass is “m.” 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 hoop 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 hoop radius “r” is much much less than “R” so that the point of contact of the hoop can be treated as if it is in the center-of-mass location of the hoop for brevity. I just had to draw the hoop out of scale so you can see it. Here h = 26 meters and R = 10 meters. Air resistance is negligible. a) What is the speed of the hoop when it reaches the top of the loop-the-loop if it is locked to the path? Show your work. b) Now we unlock the hoop from the path. Will the hoop make it all the way around the loop-theloop, or will it fall off and crash? HINT: You may want to sum to forces on the hoop at the top of the loop-theloop. Show your work. c) Now we have a skater in place of the hoop starting from rest, and the wheels of the skateboard are very, very small and light compared to the skater, and they are frictionless about their bearings and axles. The skater is unlocked from the path as well. Will the skater make it all the way around the loop-the-loop if the skater maintains balance? You may treat the skater as much shorter than the radius of the loop-the-loop here too. No, this does not make skateboarding a crime; it is SCIENCE !!! Show your work.
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 hoop ready to roll without slipping from a height “h.” The radius of the hoop is “r” and its mass is “m.”
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 hoop 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 hoop radius “r” is much much less than “R” so that the point of contact of the hoop can be treated as if it is in the center-of-mass location of the hoop for brevity. I just had to draw the hoop out of scale so you can see it. Here h = 26 meters and R = 10 meters. Air resistance is negligible.
a) What is the speed of the hoop when it reaches the top of the loop-the-loop if it is locked to the path? Show your work.
b) Now we unlock the hoop from the path. Will the hoop make it all the way around the loop-theloop, or will it fall off and crash? HINT: You may want to sum to forces on the hoop at the top of the loop-theloop. Show your work.
c) Now we have a skater in place of the hoop starting from rest, and the wheels of the skateboard are very, very small and light compared to the skater, and they are frictionless about their bearings and axles. The skater is unlocked from the path as well. Will the skater make it all the way around the loop-the-loop if the skater maintains balance? You may treat the skater as much shorter than the radius of the loop-the-loop here too. No, this does not make skateboarding a crime; it is SCIENCE !!! Show your work.
d) If the skater and hoop both make it or not make it (both crash or both not crash), why are the results the same? If the skater and the hoop results are different (one crashes and the other does not), explain why they are different.
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