Explanation and answer:
Given:
6 lb is needed to stretch 4 inches beyond natural length.
Need work done to stretch same string from natural length to 8 inches.
Solution:
string stiffness, K
= Force / stretched distance
= 6 lb / 4 inches
= 1.5 lb/inch
Work done on a string of stiffness K
= (Kx^2)/2 lb-in
= 1.5 lb/in *(8 in)^2)/2
= 48 lb-in.
Force causes motion in a body. The amount of work needed to stretch the spring 8 in. beyond its natural length is 48 lb·in.
Force is defined as the influence under which a body is in motion. It is given by the formula,
F = m x a
m = mass of the object,
a = acceleration of the object
Given to us
Force required, F = 6 lb
displacement, x = 4 in.
We know that the force for spring is given by the formula,
F = k · x
where F is the force,
k is the spring constant
x = displacement of the spring from its initial position
Substitute the values,
F = k · x
6 = k · 4
k = 1.5 lb\in
We know that work is written as the amount of force that is needed to distance an object,
W = F · dx
[tex]W = \int F dxW = k\int x dx\\\\W = k \int_0^x x dx\\\\W = \dfrac{kx^2}{2}\\\\[/tex]
As we know the spring constant of the spring now, substitute all the values,
[tex]W = \dfrac{1.5 \times 8^2}{2}\\\\W = 48\ lb\cdot in.[/tex]
Hence, the amount of work needed to stretch the spring 8 in. beyond its natural length is 48 lb��in.
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