A solid ball (sphere) rolls without slipping along a horizontal surface, then do

Question

A solid ball (sphere) rolls without slipping along a horizontal surface, then down a ramp and then along a lower horizontal surface. The ball's translational velocity (of the center of gravity) changes from ul-2.50 m/s to u2 3.70 m/s, what is the height, h, of the ramp? HINTS: The height of the center of gravity of the ball changes by a height equal to the height of the ramp. You will need to refer to Table 7.4 of the text for the moment of inertia of a solid sphere. As you work the problem, you will see that you do not need to know the mass or the radius of the bal. Make sure to enter your answer with exactly three significant figures

Explanation / Answer

Use conservation of energy.

PE_top + KE_top = PE_bottom + KE_bottom

PE_top = mgh, where "h" is the ramp's height (the value we want to find). We aren't given "m", but don't worry about that--hopefully that variable will cancel out in the end.

KE_top is a combination of the balls translational KE (=½m(v_top)²), and its rotational KE (=½I(_top)²). That is:

KE_top = ½m(v_top)² + ½I(_top)²

"I" is the ball's moment of inertia, which equals (2/5)mR² (R=ball's radius). That was probably in that table 7.4 that you wanted me to look at. Again, we don't know "R", but don't fret it right now.

Since it's rolling without slipping, "_top", the ball's initial angular velocity, is related to its linear velocity like so:

_top = v_top/R

So that means the rotational KE is:

½I(_top)² = ½(2/5)mR²(v_top/R)² = ½(2/5)m(v_top)²

(Yay! the "R" cancelled out.)

And that means the total KE_top is

KE_top = ½m(v_top)² + ½(2/5)m(v_top)² = (7/10)m(v_top)²

Now at the bottom, we have:

PE_bottom = 0

And by an logic similar to the above:

KE_bottom = (7/10)m(v_bottom)²

Now, we said PE_top+KE_top = PE_bottom+KE_bottom. SO:

mgh + (7/10)m(v_top)² = (7/10)m(v_bottom)²

Divide by "m" and it magically disappears:

gh + (7/10)(v_top)² = (7/10)(v_bottom)²

=> h = [(7/10)(v_bottom)² - (7/10)(v_top)²] / g

Given: v_top = 2.50 m/s

v_bottom = 3.70 m/s

We know : g = 9.81 m/s2

Plugging in the values:

h = [(7/10)((3.70)^2 – (7/10)(2.50)^2]/9.81

= 0.53m

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