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

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To calculate the speed of a solid sphere sliding down a frictionless incline, the relevant equation is mgh = 1/2mv^2, as there is no rotational kinetic energy involved when the sphere slips without rolling. Given the sphere's height of 1.8m and the incline angle of 22 degrees, the gravitational potential energy converts entirely into translational kinetic energy. The moment of inertia for a solid sphere is provided, but it is not needed for this scenario since the sphere does not roll. The key takeaway is that the speed can be calculated using the height and mass of the sphere, focusing solely on translational motion. Understanding the difference between rolling and slipping is crucial for applying the correct formulas.
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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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