Question #c1aad

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Question #c1aad

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Answer 1 · 2016-05-16T18:11:33

$5 β - 4 α$ $5beta-4alpha$

Explanation:

Assuming the spring obeys Hooke's law:
$F = k x$ $F=kx$

Where
$F$ $F$ =force
$k$ $k$ =spring constant
$x$ $x$ = extension

The question implies that $α$ $alpha$ and $β$ $beta$ is the total length of the spring, rather than the extension. If we define the length of the spring without any weights on it as $y$ $y$ , then

We know that
$4 = k (α - y)$ $4=k(alpha-y)$ .......equation (1)
and
$5 = k (β - y)$ $5=k(beta-y)$ .......equation (2)

If we divide the two equations:
$\frac{5}{4} = \frac{β - y}{α - y}$ $5/4=(beta-y)/(alpha-y)$
We can expand and solve for y

$5 (α - y) = 4 (β - y)$ $5(alpha-y)=4(beta-y)$
$5 α - 5 y = 4 β - 4 y$ $5alpha-5y=4beta-4y$
$5 α - 4 β = y$ $5alpha-4beta=y$ ..........equation (3)

For the 9N weight, of total length, say $z$ $z$ :
$9 = k (z - y)$ $9=k(z-y)$

If we divide this by equation (2)

$\frac{9}{5} = \frac{z - y}{β - y}$ $9/5=(z-y)/(beta-y)$

$9 β - 9 y = 5 z - 5 y$ $9beta-9y=5z-5y$
$9 β - 4 y = 5 z$ $9beta-4y=5z$
Substitute in for y from equation (3):

$9 β - 4 (5 α - 4 β) = 5 z$ $9beta-4(5alpha-4beta)=5z$
$9 β - 20 α + 16 β = 5 z$ $9beta-20alpha+16beta=5z$
$25 β - 20 α = 5 z$ $25beta-20alpha=5z$
$z = 5 β - 4 α$ $z=5beta-4alpha$

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Question #c1aad

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