Linear speed and rotational quantity

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To determine the linear speed of a hula hoop with a total kinetic energy of 0.15 J, both translational and rotational kinetic energy must be considered. The relationship between linear speed and rotational speed is defined by the condition of rolling without slipping, which indicates that the linear speed is equal to the radius multiplied by the angular velocity. The total kinetic energy combines both translational (1/2 mv^2) and rotational (1/2 Iω^2) components. Understanding these relationships and applying the appropriate formulas will yield the required linear speed. The discussion emphasizes the need for clarity in linking linear and rotational dynamics in rolling objects.
baylorbelle
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What linear speed must a 6.0×10−2 hula hoop have if its total kinetic energy is to be 0.15 J ? Assume the hoop rolls on the ground without slipping.


So, i know that the formula for linear momentum is p=mv. however, the hoop is circular, meaning it has to have a rotational velocity (omega). I can't seem to figure out how the Joules play into any equation that helps link linear and rotational speeds. Anyone know of any formulas I can use for this problem?
 
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If something is both translating and rotating, how do you determine its total kinetic energy?

What does "rolling without slipping" tell you about how translational speed relates to rotational speed?
 
if something is rolling without slipping, it has rotational speed but not translational speed?
 
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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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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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