# Determining Work and Fluid Force

## Key Questions

• The answer is $588 J$.

$W = {\int}_{a}^{b} F \left(x\right) \mathrm{dx}$

Sometimes, the simple problems are the hardest because it looks too easy so we tend to add unnecessary things. Since this is a vertical lift, we are dealing with gravity which is 9.8 $\frac{m}{s} ^ 2$. So,

$F \left(x\right) = 9.8 \frac{m}{{s}^{2}} \cdot 40 k g = 392 N$

Remember that work is force times distance. We have force which is just a constant function. And we have distance which is $\mathrm{dx}$. The tendency is to add an $x$ into $F \left(x\right) = 392 N$, but that would be incorrect. Now, let's put it together:

$a = 0$
$b = 1.5$
$W = {\int}_{0}^{1.5} 392 \mathrm{dx}$
$= 392 x {|}_{0}^{1.5}$
$= 588 J$

• The answer is $\frac{27}{4}$ ft-lbs.

Let's look at the integral for work (for springs):

$W = {\int}_{a}^{b} k x \setminus \mathrm{dx} = k \setminus {\int}_{a}^{b} x \setminus \mathrm{dx}$

Here's what we know:

$W = 12$
$a = 0$
$b = 1$

So, let's substitute these in:

$12 = k {\left[\frac{{x}^{2}}{2}\right]}_{0}^{1}$
$12 = k \left(\frac{1}{2} - 0\right)$
$24 = k$

Now:

9 inches = 3/4 foot = $b$

So, let's substitute again with $k$:

$W = {\int}_{0}^{\frac{3}{4}} 24 x \mathrm{dx}$
$= \frac{24 {x}^{2}}{2} {|}_{0}^{\frac{3}{4}}$
$= 12 {\left(\frac{3}{4}\right)}^{2}$
$= \frac{27}{4}$ ft-lbs

Always set up the problem with what you know, in this case, the integral formula for work and springs. Generally, you will need to solve for $k$, that's why $2$ different lengths are provided. In the case where you are given a single length, you're probably just asked to solve for $k$.

If you are given a problem in metric, be careful if you are given mass to stretch or compress the spring vertically because mass is not force. You will have to multiply by 9.8 $m {s}^{- 2}$ to compute the force (in newtons).

• If $F \left(x\right)$ denotes the amount of force applied at position $x$ and it moves from $x = a$ to $x = b$, then the work $W$ can be found by

$W = {\int}_{a}^{b} F \left(x\right) \mathrm{dx}$.

I hope that this was helpful.