Question

IF 5.4 J of work are needed to stretch a spring from 17 cm to 23 cm and 9 J are needed to stretch it from 23 cm to 29cm, what is the natural length of the spring.

Answer 1

I know Hooke's Law and I got
\[\int\limits_{0}^{.06} kxdx=5.4\] which turns out to be k=30

but I'm not sure where to go from here.

Answer 2

i have a theory ...do you know the answer by chance?

Answer 3

no its for homework

Answer 4

ok in Hooks Law
F = kx , the "x" is distance from natural length
so
\[5.4 = k(x+6)\]
\[9 = k(x+12)\]
solving the system
\[k = 0.6, x = 3\]
natural length of spring is 14 cm

Answer 5

get 2nd opinion though, not expert on physics stuff

Answer 6

im gonna go with it. Seems right to me.
p.s. the reason I didn't ask you this question was to give you a break from me haha

Answer 7

haha

Answer 8

buttt since you are here find the volume generated by rotating the given region about the specified line y=6x y=0 x=1

Answer 9

I know this is my graph

Answer 10

is my integral \[\int\limits_{0}^{6}\pi 6x\]

Answer 11

what is it rotating around? x=1 ?

Answer 12

http://www.webassign.net/waplots/3/4/28f75eee007175cbb776e4e09ce214.gif

Answer 13

This is the graph actually and the question says find the volume generated by rotating the given region about the specified line. R1 about AB so yea i think its 1

Answer 14

oh ok , well since region is just a nice triangle
no calculus needed, essentially you are finding volume of a cone

Answer 15

a cone is 1/2 pi r^2 right?

Answer 16

my r being 6x

Answer 17

close
\[V = \frac{1}{3}\pi r^{2}h\]

Answer 18

oh so my h is actually 6x and r is x right?

Answer 19

"r" is radius of circle, look at base

yes now you got it

Answer 20

for cone
r = 1
h = 6

the set up using shell method is:
\[2\pi \int\limits_0^1 x(6x) dx\]

Answer 21

i got 4pi

Answer 22

oops my radius is off....r = 1-x

using volume of cone, you get 2pi

Answer 23

ohhh wait why 1-x?

Answer 24

radius is distance from axis of rotation (x=1)
so when x=1, the radius should be 0
when x=0, radius should be 1

Answer 25

oh ok i see

Answer 26

so yea i got the 2pi

Answer 27

ok one more and then I'm off to bed and I will leave you alone. Till tomorrow ;)

Find the volume V generated by rotating the given region about the specified line.
R3 about OC
http://www.webassign.net/waplots/c/7/5b24e5fe28d5d47788e1c5399cf12f.gif

Answer 28

I wanna use the cylindrical shell method so i know i will have \[2\pi \int\limits_{?}^{?}rhdx\]

Answer 29

yes good
do you know what the limits will be?

Answer 30

I wanna say from 0 to 3

Answer 31

integrating wrt "x" not "y"

Answer 32

but its around the y=axis

Answer 33

true but we are using shell method

if using disc method then in terms of "y"

Answer 34

so shells and disc method are not the same limits?

Answer 35

no disc method you are summing up circle cross-sections perpendicular to axis
shell method you sum up surface areas parallel to axis

that site i sent you has some visual diagrams illustrating both methods

Answer 36

ok i was just asking bc we had another homework question that was just like this and they said do both limits and i used the same limits

Answer 37

do both methods*

Answer 38

anyway, look at radius, it goes out horizontally from vertical axis
that is clue everything in terms of "x"

Answer 39

ok so from 0 to 1

Answer 40

yes

Answer 41

you are looking at the yellow part of the graph correct?

Answer 42

yes

Answer 43

ok just checking lol

Answer 44

\[2\pi \int\limits_{0}^{1} x(3\sqrt[4]{x} -3x)\]