Answer to Question #7254 in Calculus for LeighAnne Manwiller

Question

Suppose that 2 Joules of work is needed to stretch a spring from its natural length of 45 cm to a length of 66 cm. How much work is needed to stretch it from 60 cm to 74 cm? Your answer must include the correct units.

Answer 1

Suppose that 2 Joules of work is needed to stretch a spring from its natural length of 45 cm to a length of 66 cm. How much work is needed to stretch it from 60 cm to 74 cm? Your answer must include the correct units.

Let's make the following denominations:
L1 = 45 cm (natural length)
L2 = 66 cm
A1 = 2 J
L3 = 60 cm
L4 = 74 cm

We need to find the stiffness of the spring:
x1 = L2-L1 = 66 - 45 = 21 cm;
A1 = k*x1²/2 ==> k = 2*A1/x1² = 2*2/21² = 4/121.

Let's find the work needed to stretch a spring from 45 cm to 60 cm:
x2 = L3-L1 = 60 - 45 = 15 cm
A2 = k*x2²/2 = 4/121 * 15² / 2 = 450/121 J.

Let's find the work needed to stretch a spring from 45 cm to 74 cm:
x3 = L4-L1 = 74 - 45 = 29 cm
A3 = k*x3²/2 = 4/121 * 29² / 2 = 1682/121 J.

So, the work needed to stretch a spring from 60 cm to 74 cm is
A = A3 - A2 = 1682/121 - 450/121 = 112/11 J.