Angular speed question!? I'll pick a best answer TODAY!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question

When Alicia and Zoe ride a carousel, Alcia always selects a horse on the outside row, whereas Zoe prefers the row closest to the center. These rows are 19 ft 3 in and 13 ft 11 in from the center, respectively. The angular speed of the carousel is 2.4 revolutions per minute. What is the difference, in miles per hour, in the linear speeds of alicia and Zoe?

I got something like 2.193, but the book says it is .92. Help please! I'll pick a best answer TODAY!!!!!!

Hey buddy, I can't read your answer because it gets cut off. :(

Answer 1

C=πd
Alicia = 2*19.25π'*2.4rev/min*60/5280mi...
Zoe = 2*(13 11/12)π'*2.4rev/min*60/5280min...

Alicia = 3.30 miles/hour
Zoe = 2.38miles/hr
difference =3.3-2.38=.92 mph

Answer 2

The difference in the linear speeds is 0.9139... mph, which rounded, is 0.91 mph, NOT 0.92 mph. So your book is wrong in the second significant figure.

There is NO POINT in working out their two INDIVIDUAL linear speeds. I'll work DIRECTLY with their difference. That difference is

ω Δr ft/s,

where ω is the angular speed (in radians/s), and Δr is the difference in their distances from the axis in feet.

ω = 2.4 x 2π / 60 rad/s, and Δr = 5 ft 4 ins = 5.3333... ft.

Finally, we need to convert from ft/s to mph. The most convenient way to do this is to use the conversion factor that you will find in most driving handbooks:

60 mph = 88 ft/s, so that 1 ft/s = 60/88 mph.

So, putting this all together, the difference in the linear speeds is:

2.4 x 2π / 60 x 5.3333... x 60/88 mph = 0.9139... mph, that is 0.91 mph when rounded to two significant figures.

The immediately preceding answer, and evidently the book's answer, also, illustrate the problems with retaining insufficient numerical accuracy in a calculation prior to taking the limited accuracy of the inputs into consideration.

One should always carry at least one more figure, and preferably two more, than the rounding ultimately desired, PARTICULARLY if the problem hinges on the DIFFERENCE(S) in any inputs. In this particular question, there is such a trap. As mentioned above, the key factor required is the DIFFERENCE in the two radius values, which is quite small compared to either individual radius.

(The accuracy-limiting information given above is both (i) 2.4 revs/min, and (ii) the separation of the two rows of horses on the carousel. While it might appear that the radii are given to 3 sig. figs., their DIFFERENCE of 5 ft 3 ins (or 63 inches) --- the key length input --- is only known at best to 2 sig. figs. So ultimately, only a 2 sig. fig. answer is justified. Standard calculating practice requires carrying more significant figures than the number that will appear in the final result --- and TO HELL with the two "thumbs down" ! )

Live long and prosper.

[EDIT --- 3 days later: "Hey, buddy" --- you have a VERY STRANGE INTERPRETATION of the meaning of the word "TODAY!"]

Answer 3

chasrmk is 100% correct. But, for ease in solving, I'd set the problem up as:

D = V - v = w(R - r); where w = 2.4 rev/min, R = 19.25 and r = 13 11/12. V and v are the respective tangential speeds; D is that difference you are looking for.

This saves having to set up two equations and duplicating some calculational steps.

Answer 4

The circumference of a circle is found by (pi) times diameter or 2 times (pi) times radius.
Outer radius is 19ft. 3in. call it r1
Inner radius is 13ft. 11in. call it r2
Difference in diameter is 2 (pi) r1 - 2 (pi) r2
or 2 (pi) (r1-r2)
Number of revolutions per hour is 2.4 x 60 = 144
Difference in distance travelled in 1 hour is
2.(pi).(r1-r2) .144
Work out (r1-r2) in inches and then multiply out. Divide the answer by 63,360 (assuming that a US mile is the same as a statute mile) and that gives the difference in mph.
I'm not going to do the maths for you. It is homework, after all.