How High Must a Cylinder Roll to Loop-the-Loop?

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SUMMARY

The discussion focuses on calculating the minimum height \( h \) required for a solid cylinder of radius \( r_1 \) and mass \( m \) to successfully complete a loop-the-loop of radius \( r_2 \). The key point is that at the top of the loop, the cylinder must have sufficient potential energy, expressed as \( 2r_2mg \), to maintain motion without slipping. The relationship between linear velocity \( v \) and angular velocity \( \omega \) is critical, where \( v = r_1 \omega \). The conservation of energy equation \( mgh = 2r_2mg + KE \) must be utilized to derive the necessary height \( h \).

PREREQUISITES
  • Understanding of conservation of energy principles in physics
  • Familiarity with rotational dynamics, specifically the moment of inertia
  • Knowledge of kinematics, particularly the relationship between linear and angular motion
  • Ability to analyze free body diagrams for forces acting on objects
NEXT STEPS
  • Study the conservation of energy in rolling motion
  • Learn about the moment of inertia for solid cylinders
  • Explore the dynamics of circular motion and forces at the top of loops
  • Practice drawing and analyzing free body diagrams in various scenarios
USEFUL FOR

This discussion is beneficial for physics students, educators, and anyone interested in understanding the mechanics of rolling motion and energy conservation in circular paths.

GemmaN
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In this problem you will consider the motion of a cylinder of radius r_1 that is rolled from a certain height h so that it "loops the loop," that is, rolls around the track with a loop of radius r_2. The cylinder rolls without slipping.
It looks like your standard matchbox car loop.

I need to find the minimum height h that will allow a solid cylinder of mass m and radius r_1 to loop the loop of radius r_2. "Express h in terms of the radius r_2 of the loop."

I am not quite getting a certain portion of this. I know the important part of this problem is when the cylinder is at the top of the loop.
Now, at this point, I know it has a potential energy of 2*r_2*m*g
To stay on the track without slipping, I think I need v = R*omega
At the top part of the track, the cylinder is upside down, and has no normal force... so weight matters?

I may need to use this: K = 1/2Mv_cm^2 + 1/2 I_cm*omega^2

mgh = (2)r_2(mg) + KE? + ?

I am a bit confused at this point. How am I suppose to put this together?
 
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You are correct in that the "important part of this problem is when the cylinder is at the top of the loop".

Hint -- draw a free body diagram for the system at that location.
 

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