Rotational Inertia question, keep getting wrong answer?

Question

A bowling ball made for a child has half the radius of an adult bowling ball. They are made of the same material (therefore have same mass) By what factor is both the mass and rotational inertia of the childs ball reduced compared to the adult ball? The answers is that the mass is reduced by a factor of 8 and the rotational inertia is reduced by a factor of 32. I keep coming out with 23/20

Lamar Hirthe · Accepted Answer

Not clear how you conclude the two have the "same mass." They don't if they're made of the same material and have the same density. In which case the mass of the child's ball is m = rho v and of the adult's is M = rho V, where V = 4/3 pi R^3 > 4/3 pi r^3 = v. Then V/v = (R/r)^3 and V = v (R/r)^3 is true. R is the adult radius, r is the kid's ball. rho is the mass density of both balls.When r = (1/2)R, we have V = 2^3 v = 8v for this case. Then M/m = rho V/rho v = V/v = 8v/v = 8. So M = 8m which clearly shows the mass of the adult ball is 8 times that of the kid's. Or m = M/8 QED.The two I's are I = kMR^2 and i = kmr^2; so i/I = kmr^2/kMR^2 =(1/8)(R/2//R)^2 = (1/8)(1/4) = 1/32 and thus i = (1/32)I QED. The k's are the same value and cancel out in the ratio because the balls are the same shape and density distribution.As you didn't show your work, I have no clue where you erred. But the textbook answers are correct. For your information, when comparing like factors (like the two masses and the two inertia) use ratios, like M/m and I/i, to solve the problems. Most of the time a lot of variables will cancel out leaving only a couple to work with.

Lamar Hirthe · Answer

Volume of a sphere = (4/3) pi R3.Mass = density times volume (we suppose constant and equal densities)As radius adult = 2 radius child; volume adult = 8 times volume child. Therefore Mass adult = 8 times Mass child.Moment of inertia of a sphere = (2/5) M R2MI child = (2/5) (1/8 M adult) (R adult / 2)2 = (2/5) Madult Radult2 (1/8)*(1/4) = MI adult / 32

Lamar Hirthe · Answer

For #2, it particularly is each and every of the forces. not some thing fairly "opposes" the centripetal stress. the strain is unopposed and takes result - it reasons acceleration. the value does not replace, in spite of the undeniable fact that the value variations because of actuality the path variations. you acquire #3 maximum suggestions-blowing. The 4.5N threw me off too. in case you paintings it out 4.5N is the centripetal stress you would be able to assume.

Prof. Carmelo Hilpert

Rotational Inertia question, keep getting wrong answer?

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