How to Calculate the Speed of a Solid Sphere on a Frictionless Incline?

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SUMMARY

The discussion focuses on calculating the speed of a solid sphere with a radius of 20 cm sliding down a frictionless incline at a 22-degree angle, starting from a height of 1.8 m. The moment of inertia for the sphere is given as 2/3MR². The relevant equation used is mgh = 1/2Iw² + 1/2mv², which indicates that when the sphere slips without rolling, only translational kinetic energy is considered, eliminating the rotational component. The solution emphasizes the distinction between rolling and slipping scenarios in kinetic energy calculations.

PREREQUISITES
  • Understanding of gravitational potential energy (mgh)
  • Familiarity with kinetic energy equations (1/2mv²)
  • Knowledge of moment of inertia for a solid sphere (2/3MR²)
  • Basic principles of rotational motion and angular velocity
NEXT STEPS
  • Study the derivation of the moment of inertia for various shapes
  • Learn about energy conservation in mechanical systems
  • Explore the differences between rolling and slipping motion
  • Investigate the effects of friction on motion dynamics
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Students in physics, particularly those studying mechanics, as well as educators and anyone interested in understanding the dynamics of objects on inclines.

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Homework Statement



A solid sphere of radius 20cm is positioned at the top of an incline that makes 22 degrees angle with the horizontal. This initial position of the sphere is a vertical distance 1.8m above its position when at the bottom of the incline. Moment of inertia of a sphere with respect to an axis through its center is 2/3MR^2. Calculate the speed of the sphere when it reaches the bottom of the incline in the case where it slips frictionlessly without rolling.

Homework Equations



mgh = 1/2Iw^2 + 1/2mv^2

The Attempt at a Solution



I know how to calculate the speed when the object rolls down without slipping, but what do I do if it does? if there a formula? Thank you for your help!
 
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If it slips without rolling then there is no rotational KE, just translational KE.

If it rolls without slipping then there is both rotational and translational KE. The are related by the condition for "rolling without slipping", which is: v = \omega r.
 
Thank you so much!
 

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