If $4 \mathrm {~J}$ of work are needed to stretch a spring from $10 \mathrm {~cm}$ to $12 \mathrm {~cm}$ and another $20 \mathrm {~J}$ are needed to stretch it from $12 \mathrm {~cm}$ to $14 \mathrm {~cm}$, what is the natural length of the spring? Round the answer to nearest integer.
Select the correct answer.
A) $12 \mathrm {~cm}$
B) $8 \mathrm {~cm}$
C) $13 \mathrm {~cm}$
D) $11 \mathrm {~cm}$
E) $10 \mathrm {~cm}$

Question

Quizplus · Accepted Answer

Let the natural length of the spring be $x$. Then, using Hooke's Law, we have that the work required to stretch the spring from $x$ to $x+2$ is given by
$$W = \int_{x}^{x+2} F(x') dx' = \int_{x}^{x+2} k(x'-x) dx' = \frac{k}{2} (x+2)^2 - \frac{k}{2} x^2$$
where $F(x')$ is the force required to stretch the spring to length $x'$, and $k$ is the spring constant.

Using the given information, we can set up two equations:
$$\frac{k}{2} (12)^2 - \frac{k}{2} (10)^2 = 4$$
$$\frac{k}{2} (14)^2 - \frac{k}{2} (12)^2 = 20$$
Simplifying each equation and subtracting the first from the second yields
$$\frac{k}{2} (14^2 - 12^2) = 16$$
$$\frac{k}{2} \cdot 100 = 16$$
$$k = \frac{8}{25}$$

Substituting $k$ back into the first equation will give us an equation in $x$:
$$\frac{1}{50} (12)^2 - \frac{1}{50} x^2 = 4$$
Solving for $x$ gives us $x = 11.2$, which rounded to the nearest integer is $\boxed{\textbf{(D) } 11}$.

If 4 J4 \mathrm {~J}4 J Of Work Are Needed to Stretch a Spring From

If $4 \mathrm {~J}$ Of Work Are Needed to Stretch a Spring From