| Well, here goes. Let l , l be the lengths of the arms, and let w , w be
1 2 1 2
the weights of the pans on those arms, respectively. Then,
w l + X l = w l + X l , X = A, B, C
1 1 1 2 2 1 2
w l + X l = w l + X l , X = A, B
1 1 2 1 2 2 2
Let k = (w l - w l )/l , and l = l /l + 1. Then,
1 1 2 2 1 2 1
k + X = X (l-1), X = A, B, C
1
k + X = X (l-1), X = A, B
2
So, 2
k(l-1) + X(l-1) = X (l-1) = k(l-1) + k + X = k + X , X = A, B
1 2 2
C = C (l-1) - k
1
But 2 2
k = A (l-1) - A = B (l-1) - B
1 2 1 2
and
2
(A -B )(l-1) = (A -B )
1 1 2 2
So,
A - B A B - A B
2 2 1 2 2 1
C = C sqrt(------) + -----------
1 A - B A - B
1 1 1 1
|
| Re .1:
Sorry, but I think you made a mistake. Consider the line
2
k(l-1) + X(l-1) = X (l-1) = k(l-1) + k + X = k + X , X = A, B.
1 2 2
The penultimate expression is k(l-1) + k + X2. This simplifies to
k l + X2, but you have omitted the l in the last expression.
-- edp
|
| The balance problem seemed a lot harder in high school. Maybe I did learn
something in college.
Suppose a weight x on the left balances a weight y on the right. Then the
center of gravity of the arms, pans, and weights is at the "center" of the
balance. Thus,
(x+wl)*ll-(y+wr)*lr = 0.
where ll is the length of the left arm and wl is the weight of the left pan
plus an adjustment for the weight of the left arm, and lr and wr are the
same quantities for the right side.
The five given balancing situations are represented by
(A+wl)*ll-(A1+wr)*lr = 0, (0)
(B+wl)*ll-(B1+wr)*lr = 0, (1)
(C+wl)*ll-(C1+wr)*lr = 0, (2)
(A2+wl)*ll-(A+wr)*lr = 0, and (3)
(B2+wl)*ll-(B+wr)*lr = 0. (4)
Solving (0) and (3) for A gives
(A1+wr)*lr/ll-wl = A = (A2+wl)*ll/lr-wr. (5)
Similarly, (1) and (4) give
(B1+wr)*lr/ll-wl = B = (B2+wl)*ll/lr-wr. (6)
(2) gives
(C1+wr)*lr/ll-wl = C. (7)
Let l = lr/ll. Then
(A1+wr)*l-wl = (A2+wl)/l-wr, (from (5)) (8)
(B1+wr)*l-wl = (B2+wl)/l-wr, and (from (6)) (9)
(C1+wr)*l-wl = C. (from (7)) (10)
Expanding these gives
A1*l+wr*l-wl = A2/l+wl/l-wr, (11)
B1*l+wr*l-wl = B2/l+wl/l-wr, and (12)
C1*l+wr*l-wl = C. (13)
Multiplying (11) and (12) by l gives
A1*l^2+l*(wr*l-wl) = A2+wl-wr*l and
B1*l^2+l*(wr*l-wl) = B2+wl-wr*l.
Let w = wr*l-wl. Then
A1*l^2+l*w = A2 - w,
B1*l^2+l*w = B2 - w, and
C1*l+w = C. (14)
Solving for w gives
(A2-A1*l^2)/(l+1) = w = (B2-B1*l^2)/(l+1). (15)
Thus
A2-A1*l^2 = B2-B1*l^2.
l^2 = (A2-B2)/(A1-B1).
l = sqrt((A2-B2)/(A1-B1)).
Substituting this into (15) gives
w = (A2-A1*(A2-B2)/(A1-B1))/(1+sqrt((A2-B2)/(A1-B1))).
Putting l and w in (14) gives
C = C1*sqrt((A2-B2)/(A1-B1)) +
(A2-A1*(A2-B2)/(A1-B1))/(1+sqrt((A2-B2)/(A1-B1))).
When this expression is "simplified", the result is
A1*B2+A2*B1
C = - ----------- +
A1-A2-B1+B2
A1*B2+A1*C1+A2*B1-A2*C1-B1*C1+B2*C1
----------------------------------------- * sqrt(A1*A2-A1*B2-A2*B1+B1*B2).
A1^2-A1*A2-2*A1*B1+A1*B2+A2*B1-B1^2-B1*B2
I am willing to bet that there is a mistake somewhere in there. Can you
imagine the poor judges who had to grade a hundred of these, all handwritten
in a hurry?
-- edp
|