Roling vs Sliding Down an incline

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Rolling objects accelerate faster down an incline compared to sliding objects due to the conversion of gravitational potential energy into both linear and angular kinetic energy. The rolling object gains energy from both its motion and rotation, while the sliding object primarily converts potential energy into linear kinetic energy, with friction affecting its speed. The equations of motion for rolling objects differ based on their shape, such as balls and cylinders, and involve factors like the incline angle and moment of inertia. For sliding objects, the kinetic energy can be expressed in relation to the coefficient of friction and the incline angle. Understanding these dynamics is crucial for analyzing the energy transformations in both scenarios.
Kalie
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Think about 2 objects (one rolling one sliding) down an incline. In terms of energy, see what you came up with and how they differ...be verbose!

All right I know that the rolling object would go down the incline faster and...i'm not sure but I think...the sliding one would gather the most energy do to friction but I don't know how to prove this...any clues?
 
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The rolling one will gain both linear and angular kinetic energy, right? Can you show the equations for these? Can you write the equation of motion for a rolling object (ball and cylinder will be slightly different) as a function of the angle of the incline?

What is the equation of motion for the sliding object versus mu and the angle of the incline? What is the KE of the sliding object versus time?
 
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