Determining Work and Fluid Force
Key Questions

The answer is
#588J# .Always start with the definition:
#W=int_a^b F(x)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
#m/s^2# . So,#F(x)=9.8m/(s^2)*40kg=392N# Remember that work is force times distance. We have force which is just a constant function. And we have distance which is
#dx# . The tendency is to add an#x# into#F(x)=392N# , but that would be incorrect. Now, let's put it together:#a=0#
#b=1.5#
#W=int_0^(1.5) 392 dx#
#=392x_0^(1.5)#
#=588J# 
The answer is
#(27)/4# ftlbs.Let's look at the integral for work (for springs):
#W=int_a^b kx \ dx = k \ int_a^b x \ dx # Here's what we know:
#W=12#
#a=0#
#b=1# So, let's substitute these in:
#12=k[(x^2)/2]_0^1#
#12=k(1/20)#
#24=k# Now:
9 inches = 3/4 foot =
#b# So, let's substitute again with
#k# :#W=int_0^(3/4) 24xdx#
#=(24x^2)/2_0^(3/4)#
#=12(3/4)^2#
#=(27)/4# ftlbsAlways 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
#ms^(2)# to compute the force (in newtons). 
If
#F(x)# 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(x)dx# .
I hope that this was helpful.
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