Question

Suppose the work required to stretch a spring from 1.1m beyond its natural length to a meters beyond its natural length is 5 J, and the work required to stretch the spring from a meters beyond its natural length to 4.8 m beyond its natural length is 9 J. Find the spring constant k and the value of a (approximate answers acceptable).

Answer 1

Answer:

Explanation:

The formula for the work done on a spring is:

$W = \frac{1}{2}kx^{2}$

where k is the spring constant and x is the change in length of the string

For the first statement,

x = (a-1.1), W = 5

=> $5 = \frac{1}{2}k(a-1.1)^{2}$

Now making k the subject of formula, we have:

$k = \frac{10}{(a-1.1)^2} ---------------- (A)$

For the second statement,

x = (a-4.8), W = 9

=> $9 = \frac{1}{2}k(a-4.8)^{2}$

Now making k the subject of formula, we have:

$k = \frac{18}{(a-4.8)^{2} } ------------------------- (B)$

Equating A and B since k is constant, we have:

$\frac{10}{(a-1.1)^{2} } = \frac{18}{(a-4.8)^{2} }$

solving for the value of a

$8a^{2} +56.4a-208.62=0$

solving for a, we get:

a = 2.6801 or -9.7301

but since length cannot be negative, a = 2.68m

substituting the value of a in equation B, we have:

$k=\frac{18}{(2.68-4.8)^{2} }$

k = 4.005