[3rd repost] Help with physics prob?

Question

Small coin is sliding inside a cone without friction. The axis
of the cone is vertical. Minimum distance from the tip of
the cone Dmin = 30cm, and maximum distance Dmax = 90cm.

What is average by time distance from the tip
of the cone to the coin?

[a] 50cm
[b] 55cm
[c] 60 cm
[d] 65 cm

In previous versions I also listed

[e] none of the above,

but gradually gave up hope on this one.

I will prove tomorrow, that 65 is exact.

There are three equations
conservation of angular momentum L,
conservation of energy E = K+P, and
virial of forces:

Energy & angular momentum:
(L/Rmin)²/2 + gHmin = E
(L/Rmax)²/2 + gHmax = E

gHmax Rmax² - gHmin Rmax² = E
(Rmax² - Rmin²)

P + K = E =
g(Rmax³ - Rmin³)/(Rmax² - Rmin²) tg(a)

Virial:
2 = = 2

Finally:
2/3 E = = gsin(a) =
g(Dmax³ - Dmin³)/(Dmax² - Dmin²)sin(a)

=
2/3 (Dmax³ - Dmin³)/(Dmax² - Dmin²) =
65 cm

**************
The virial part perhaps needs elaborating:

2 = - virial of (N + mg) with respect to
the tip of the cone = 0 + mgh = P

Answer 1

I'm interpreting the problem as asserting that one of the four numbers listed is actually even though there's insufficient information to calculate it otherwise (at least I'm pretty sure there isn't). That might explain why the problem is phrased as a multiple choice question rather than simply asking for an answer.

Following Bekki's suggestion from the previous, express dr/dt in terms of r alone using energy conservation:

dr/dt = sqrt(2E - 2agr - a^2L^2/r^2)

where a is the sine of the angle of the cone, L is the angular momentum.

The expression inside the radical is (1/r^2) times a cubic polynomial in r so it can be written:

dr/dt = (1/r)sqrt(2Er^2 - 2agr^3 - a^2L^2)
= (K/r)sqrt((r+c)(r-d1)(d2-r))

where K is a constant and d1, d2, and -c are the three roots of the polynomial. We know that d1=30, d2=90 and c is some *positive real* number.

-->

dr/dt =
K sqrt(1/r+c/r^2) sqrt((r-d1)(d2-r))
= K f(r) g(r)

where f(r) is a decreasing function of r and g(r) is symmetric about r = (1/2)(d1+d2) = 60.

Since the average of r is weighted by 1/(dr/dt) (when averaged or integrated over r), it follows that must be strictly greater than 60 for all values of c.

--> a,b,c are impossible
--> answer is d (65)

*****Edit******

I see. c is determined by the fact that the polynomial has no r term:
-->
c = d1*d2/(d2+d1) = 45/2
-->
dt/dr = r/sqrt((r+45/2)(r-30)(90-r))

= I2/I1

where:

I2 = \integral_30^90 r^2 dr/sqrt((r+45/2)(r-30)(90-r))
I1 = \integral_30^90 r^1 dr/sqrt((r+45/2)(r-30)(90-r))

That does appear to give 65 numerically - there's probably some simple way to calculate it analytically. I suppose there's some vastly easier way to see this.

Answer 2

None of your given information has anything to do with time, so it is not possible to take any kind of a time based average. The average distance is (90+30)/2 = 60 cm.

Answer 3

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