The centrifugal force acting on a mass traveling in a circular path varies directly as the square of its speed and inversely as the radius of its circle. If a 1200 pound force is acting on a mass traveling 40 feet per second along a path that is 6 feet in diameter, what speed in feet per second will the same mass travel on the same path with a 90 pound force acting on it? Please show work!

Question

The centrifugal force acting on a mass traveling in a circular path varies directly as the square of its speed and inversely as the radius of its circle.  If a 1200 pound force is acting on a mass traveling 40 feet per second along a path that is 6 feet in diameter, what speed in feet per second will the same mass travel on the same path with a 90 pound force acting on it? Please show work!

jim_thompson5910 · Answer

Step 1) Translation

How would you translate "The centrifugal force acting on a mass traveling in a circular path varies directly as the square of its speed and inversely as the radius of its circle"

anonymous · Answer

y = kx^2
y = k/x?

jim_thompson5910 · Answer

good, but I should backtrack a bit

jim_thompson5910 · Answer

Step 1) Identity the variables

Let 

F = force (or centrifugal force)

s = speed

r = radius

jim_thompson5910 · Answer

then step 2) Translation


""The centrifugal force acting on a mass traveling in a circular path varies directly as the square of its speed" ----> F = ks^2


and


"The centrifugal force acting on a mass traveling in a circular path varies... inversely as the radius of its circle" ---> F = k/r


So combine the two to get


F = (ks^2)/r

anonymous · Answer

Wait, wouldn't it be k(ks^2)/r? Since it's k/r? I'm not quite completely following when combining them.

jim_thompson5910 · Answer

basically we have F = ks^2 and F = k/r and we "combine" the ks into one, but we're kinda abusing notation because the k's are different


So we should have F = ms^2 and F = n/r where m and n are constants


Combine the two: F = (ms^2)*(n/r) = (m*n*s^2)/r = (ks^2)/r


Where k is a constant and k  =m*n


So even though the prev line of reasoning was off, it still gives us the same equation.

anonymous · Answer

Oh. ok. I get what happened now. :)

jim_thompson5910 · Answer

alright, that's great

jim_thompson5910 · Answer

So do you know where to go from here?

anonymous · Answer

Do I just plug in numbers?
1200 = mn40^2/3
3600 = 1600mn
mn = 9/4
90 = 9/4(s^2)/3
270 = 9/4(s^2)
120 = s^2
s = 2√30?

jim_thompson5910 · Answer

You can use k to make things simpler, but it will give you the same answer.

jim_thompson5910 · Answer

In either case, you get k = 9/4 (or mn = 9/4) and s = 2*sqrt(30), so very nice job

anonymous · Answer

Thank you! :)

jim_thompson5910 · Answer

Also, you might want to convert everything to decimal, so 2*sqrt(30) = 10.954451 roughly

anonymous · Answer

Oh. I'm doing this for a math team question and we're not allowed to use calculators which is why I left it in simplest radical form.

jim_thompson5910 · Answer

oh gotcha, then you got it