Question

Please help me with this calculus problem!



The position, x, of the midpoint of the vibrating violin string is described by the formula x=0.05cos(880pi(t)), where x is measured in centimeters and t is measured in seconds.

i) Determine the formulas for the velocity and the acceleration of the midpoint of the string.

ii) Determine the maximum speed of the vibration of the string in cm/s

iii) Show that the acceleration and position satisfy the differential equation, d^x/dt^2 + (880pi)^2x=0

Answer 1

Hi amistre64 :)

Answer 2

lol .. i remember YOU! :)

Answer 3

I finnaly got that other question DONE

Answer 4

yay!!

Answer 5

It was like 5 pages :O

Answer 6

Tink you could help me out here?

Answer 7

take the 1st and 2nd derivative for the first part

Answer 8

I plugged that into wolfram, I have no idea how it got that answer... http://www.wolframalpha.com/input/?i=Derivative+of+0.05cos%28880pi%28t%29%29

Answer 9

@amistre64?

Answer 10

x = 0.05 cos ( 880pi t )

x' = -0.05 sin ( 880pi t ) * 880pi

x'' = -0.05 cos ( 880pi t ) * (880pi)^2

Answer 11

Ok thanks for that, onto part ii) :D

Answer 12

lol wolfram thought you meant \(\pi(t)\) as if \(\pi\) was a function of t because you put it in parentheses

Answer 13

pi(t) is notation for a certain type of function i think :)

Answer 14

OOOHHH LOL I feel dumb...

Answer 15

http://www.wolframalpha.com/input/?i=Derivative+of+0.05cos%28880pi*t%29

Answer 16

the maximum speed is when x'' = 0

Answer 17

amistre64, isnt your answer right as well?

Answer 18

my answer is never EVER right, except on occasion when I am actually answering a different question

Answer 19

LOL

Answer 20

cos(+-pi/2) = 0 so we have to make the innards equate to +-pi/2

Answer 21

hmm... I have no idea on how to do that xD

Answer 22

its simple; say we want to know when 880pi t = +- pi/2

we would simply state

880pi t = pi/2 and solve for t

Answer 23

t = 1/1760 i think

Answer 24

Yes, I know how to do that, thats simple

Answer 25

But how would we do, -0.05 cos ( 880pi t ) * (880pi)^2 = 0?

Answer 26

the trick tho is to realize the cos(whatever) is never greater than 1

Answer 27

ok

Answer 28

the biggest value of the x' is when cos(...) = 1

Answer 29

.05 * 880pi id say

Answer 30

So what should I write for part ii)?

Answer 31

as long as im reading it right, yeah

Answer 32

when is x' the biggest value? when cos(..) is its biggest value and cos(..) biggest value is 1

Answer 33

hmm?

Answer 34

well, sin is the same beast but with left shoes on

Answer 35

Can you interpret this? http://www.wolframalpha.com/input/?i=+0%3D-0.05+cos+%28+880pi+t+%29+*+%28880pi%29%5E2

Answer 36

spose I got this assortment
5*83*925*X

if X can range from -1 to 1; when is this assortment at its biggest value?

Answer 37

1?

Answer 38

yep

so when is the speed going the fastest? when sin(...) = 1 right?

Answer 39

Yes

Answer 40

and 1 times anything doesnt change its value

Answer 41

yeah...

Answer 42

.05*880 is the fastest speed we can get with x'

Answer 43

forgot a pi ...

Answer 44

But arent we doing it for x''?

Answer 45

oo yea nvm cm/s right

Answer 46

a'' aint speed; part2 wants speed

Answer 47

So far part ii) what should I write exactly?

Answer 48

that the highest value relating to speed is when the sin junk = 1 (since thats its highest value). and we are left with .05*1*880pi

Answer 49

We are equating x' = -0.05 sin ( 880pi t ) * 880pi to zero right?
And so I should write -0.05sin(1)(880pi) = -138.22
Therefore the maximum speed of the vibration of the string is -138.22cm/s?

Answer 50

if you think 0 is a fast speed ....

Answer 51

So I am wrong?

Answer 52

your a little off in your presentation, yes

Answer 53

lol

Answer 54

sin(880pi t) = 1 when 880pi t = pi/2

but regardless of that; sin(n pi/2) = 1

Answer 55

and speed is an absolute value so drop the negative

Answer 56

So I keep my answer the same except I take the negative sign out?

Answer 57

take out the negative; or wrap it in |....|

change sin(1) to sin( n pi/2) where n stands for any odd integer value

Answer 58

But the answer is 138.22cm/s correct?

Answer 59

is that .05 * 880pi ?

Answer 60

pi as in 3.14 or 180?

Answer 61

180 is an angle measure that has no value in most problems; always use a radian measure unless directed otherwise

Answer 62

Then yes I get 138.23

Answer 63

ok

Answer 64

ONTO PART iii)!!

Answer 65

take x'' and add it to x*(880pi)^2 and see if you get zero

Answer 66

\[(-0.05\cos(880\pi t)(880\pi)^2)+(880\pi)^2\]?

Answer 67

close; you forgot the x part tho

Answer 68

x*880pi^2 = .05cos(880pit) 880pi^2
+ x'' = -.05cos(880pit) 880pi^2
---------------------------------
= 0

Answer 69

Is that what I should write?

Answer 70

do you know what all the parts are? if so, fill them in and chk

Answer 71

Fill them in with what?

Answer 72

spose you have an equation such as:

t+9 = 3

and you want to know if t=0 is a solution to the equation; what would you do?

you know the equation, you know the value of all the "parts" ; what do you do?

Answer 73

t=-6?

Answer 74

in this case you know the value of x''
you know the value of x

you know that they are asking if x'' + x*880pi^2 = 0

fill in the parts and see if its a match

Answer 75

Whats x?

Answer 76

what does the problem say x equals? quit thinking that its some independant variable for starters; read it in the context of the problem itself.

Answer 77

x is 0

Answer 78

no it isnt, re read the problem

Answer 79

for starters; if x=0, then x'' does NOT equal -.05cos(880pi t) 880pi^2

Answer 80

I have no idea

Answer 81

The position, x, of the midpoint of the vibrating violin string
is described by the formula

x = 0.05 cos( 880pi t )

Answer 82

Then what is t?

Answer 83

t is anything that it wants to be

Answer 84

usually it denotes time

Answer 85

Ok, im confused... What do we do with this ?
x*880pi^2 = .05cos(880pit) 880pi^2
+ x'' = -.05cos(880pit) 880pi^2
---------------------------------
= 0

Answer 86

we look at it and see if its true wince we are asked in part 3 if:

x'' + x*880pi^2 = 0

Answer 87

So,

Answer 88

How do we know if its true?

Answer 89

IF x''+x*880pi^2 = 0 ; then its true

IF x''+x*880pi^2 does NOT = 0 ; then its NOT true

Does x''+x*880pi^2 = 0 ??

Answer 90

How do we do figure out whether its true or not?

Answer 91

by filling in for x'' and x into the equation and determining if the result is indeed zero


x*880pi^2 = .05cos(880pit) 880pi^2
+ x'' = -.05cos(880pit) 880pi^2
---------------------------------
= 0

Answer 92

So, \[(0.05\cos(880\pi t))(x(880\pi)^2)\]

?

Answer 93

what does x'' equal?

Answer 94

-.05cos(880pit) 880pi^2

Answer 95

good

what does x*88opi^2 equal ?

Answer 96

(0.05cos(880πt))((880π)^2)?

Answer 97

yep

Answer 98

when we combine the two of them as the equation tells us to; do we get a result of zero?