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By Avery Fiore
Written Analysis
Written Analysis
Free Body Diagrams
Free Body Diagrams
There are 4 States the hula hoop is at: when the hula hoop is traveling in the air, when the hula hoop is bouncing off of the ground, when the hula hoop is contacting the ground and slipping, and when the hula hoop is rolling on the ground. (In all of the following free-body diagrams, technically there is a force from air resistance, however, because the hula hoop is so thin and from analyzing the data from the various energy and kinematic graphs, the air resistance force can be counted as being non-existent because it is so small.)
Hula Hoop Free Body Diagram (In Air)
Hula Hoop Free Body Diagram (In Air)
While in the air, the only force acting on the hula hoop is the force from gravity. This accelerates the hula hoop towards the ground (increasing the magnitude of vertical velocity) while the angular and horizontal velocities are kept the same as no forces or torques are acting on these variables.
Hula Hoop Free Body Diagram (On Ground Slipping and Bouncing)
Hula Hoop Free Body Diagram (On Ground Slipping and Bouncing)
As the hula hoop contacts the ground for a brief period of time, several new forces and a torque appear. The normal force of the hula hoop against the ground is greater than the gravitational force accelerating the hula hoop upward. This new vertical velocity for the hula hoop is smaller as the gravitational potential energy of the hula hoop eventually goes to 0. At the contact point, there is a frictional force that accelerates the hula hoop's horizontal velocity to the left, (decreasing the magnitude of the hula hoop's horizontal velocity). This force as it is away from the center of mass of the hula hoop also creates a torque. This torque accelerates the hula hoop angular velocity clockwise (also decreasing the magnitude of the hula hoop's angular velocity). However, once the hula hoop stops contacting the ground, as it moves upward, the hula hoop returns to the first free body diagram with the frictional force, normal force, and torque not existing.
Hula Hoop Free Body Diagram (On Ground Slipping)
Hula Hoop Free Body Diagram (On Ground Slipping)
The Hula Hoop bounces a few times, each time the hula hoop comes off the ground a shorter and shorter distance. This is due to friction decreasing the kinetic energy of the hula hoop. At some point, the hula hoop starts contacting the ground. Now the normal force simply cancels out the gravitational force with both magnitudes being equal meaning there is 0 vertical acceleration. And the vertical velocity of the hula hoop at this point is also effectively 0 due to kinetic and gravitational energy being lost to friction. The frictional force is still at play, accelerating the center of mass of the hula hoop to the left (decreasing the magnitude of horizontal velocity) while the hula hoop is accelerated clockwise due to the torque (also decreasing the magnitude of angular velocity).
Hula Hoop Free Body Diagram (On Ground Rolling)
Hula Hoop Free Body Diagram (On Ground Rolling)
At some point, the hula hoop stops slipping as the tangential velocity of the hula hoop has become equal to the horizontal velocity of the hula hoop. If the hula hoop is thrown with a high horizontal velocity and low angular velocity (initial angular velocity would always be positive and spin in the anti-clockwise direction), the hula hoop ends up spinning clockwise as the torque starts spinning the hula hoop clockwise before the horizontal velocity of the hula hoop decreases below 0. If the hula hoop is thrown with a low horizontal velocity and a high angular velocity, the hula hoop spins anti-clockwise, as the horizontal velocity of the hula hoop decreased below 0 due to the frictional force before the torque caused the hula hoop to start spinning clockwise. This determines which way the hula hoop ends up spinning and whether the hula hoop ends up at the left or right side of the scene.
Solving for Constants to Create Code
Solving for Constants to Create Code
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