Sphere on Incline: Kinetic Energy of a Rolling Sphere

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A solid sphere rolls down a 35° incline without slipping, starting from rest, covering a distance of 3.9 m. In a frictionless scenario, the translational kinetic energy at the bottom equals the initial potential energy, calculated using the mass and height. When rolling, part of the energy converts into rotational kinetic energy due to the sphere's moment of inertia. The final kinetic energy differs between the rolling and slipping cases, as rolling involves both translational and rotational components. The discussion highlights the importance of understanding energy conservation in different motion scenarios.
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A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 3.9 m down a q = 35° incline. The sphere has a mass M = 4.6 kg and a radius R = 0.28 m.
https://online-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?courses/phys101/fall07/homework/08/03/3.gif

Suppose now that there is no frictional force between the sphere and the incline. Now, what is the translational kinetic energy of the sphere at the bottom of the incline?
 
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I figured out that the velocity is 5.6 m/s, but I don't know what the KE tran would be at the bottom without friction.
 
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The problem statement is incomplete, but I infer that you are being asked for the translational KE at the bottom for the two cases: the ball rolls down the incline, or it just slips down the incline (frictionless case).

If it just slips, then the KE final equals the PE initial, right? Why?

If it rolls, then some energy goes into the rolling motion (look up moment of inertia). What is the equation for the rotational energy of a sphere? What does that do to the final KE of the sphere at the bottom?
 
nvm i figured it out.
 
Doh!
 

Thread 'Rolling without slipping on a curved surface'

Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
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