The Grove

matt

01:17

@bobble apparently not

bobble

apparently. and I have to leave soon, so it would be too late.

matt

hmm

i'd offer but i'm not in physics yet

Deusovi

03:11

@bobble sorry for the delay - available now but it's likely too late

bobble

I'm here, maybe you can point me in the right direction

I'll pull up the problem

Okay, so there's a stupidly complicated set-up

On table, cart is attached to spring and is in Simple Harmonic Motion

cart is also attached to a sting which extends off table and is attached to a bucket

the bucket has a pencil attached to it

There is 0.5kg in the bucket and 9.5kg in the cart

And now the problem starts

A paper is placed such that when everything is at rest, and there are no weights in bucket or cart, the pencil is at its top

When the weights are in, and the cart is in SHM, and the paper is moved horizontally at a constant rate, the pencil draws a sine curve with amplitude of 2cm, with top 3cm from top of paper

so now: at what speed did the paper move horizontally?

(period of 8cm)

Deusovi

wow you weren't kidding about "stupidly complicated setup"

bobble

I am.. um.. very stuck

oops, cart is attached to "string" off the table, typo

can you provide any pointers at all?

Deusovi

okay so

first: draw a picture. make sure you can see what's going on.

bobble

there will be slight lag as I draw picture

matt

03:18

pictures always help in my experience

Deusovi

i've got a picture that i've drawn

it's not good but it works

please enjoy the artistic masterpiece

bobble

ooh pretty

should I draw arrows on it for forces?

Deusovi

that's the usual second step!

matt

@Deusovi considerably better than this "absolutely garbage picture"

(no offense ofc)

bobble

(image editor is being annoying)

Deusovi

03:21

to be fair, for that other one i was literally just drawing "a point on two curves"

bobble

Assuming that the cart is past the halfway point

arrows not to scale

matt

ooh yay fancy shadows on the words

bobble

I don't see anywhere where they explicitly say "no friction between cart and table", but I'm assuming it

Deusovi

yeah it's probably a cart because it rolls

bobble

So, one thing: for the cart, F_norm = F_grav, since the cart isn't moving vertically

Deusovi

03:26

plus friction would mean that it's not SHM because the oscillations would get smaller and smaller

bobble

true!

Deusovi

so. what relates the horizontal speed of the paper to anything else?

bobble

horizontal speed of cart = vertical speed of pencil, since they're attached through string

so amplitude of pencil (2cm) is amplitude of cart's SHM

Deusovi

that's true but not the question i asked

bobble

oh I read "cart"

Deusovi

03:28

though starting from there is perfectly fine - we can do that too

bobble

horizontal speed of paper is related to period?

because it moves 8cm / period

Deusovi

sounds good to me! it's the only thing i see that's related to the horizontal speed of the paper

bobble

so now we need to find period

Deusovi

yeah, that seems like a more approachable goal, and would let us use SHM stuff

bobble

how does the 0.5kg in the bucket affect the SHM of the cart?

since then there is a force from something that's not the spring

that was what I was getting stuck on

I do have T = 2pi * sqrt(m/k), but I don't know T or k, and I'm not sure what mass to use for m - cart mass? cart mass + bucket mass? something else?

Deusovi

03:32

well, let's figure it out!

we can't just use that equation directly here - it's not just a single object on a spring

(though it may be useful later)

bobble

something that might help: when the system is at rest, the pencil is 5cm below where it was without any extra weights

Deusovi

yep, that's very good to know

bobble

If pencil is 5cm below, that means cart's middle position is 5cm out

So the starting U_s would have x increase by 5cm

and the bucket would have U_g decrease because h decreases by 5cm

does any of this help?

Deusovi

U is... potential energy?

bobble

yes, U_s = spring potential energy, U_g = gravitational potential energy

Deusovi

03:36

ah alright

bobble

that's what my teacher uses

I have no idea why

Deusovi

right, i think that's what we used back when i took physics too, but just forgot the variable name

well obviously it's u for "uotential"

bobble

unintentionally confusing?

Deusovi

alternatively: because P stands for "momentum"

Ankoganit

Makes perfect sense

Deusovi

03:39

anyway! hooke's law says that the restoring force is proportional to the amount of displacement

matt

@Deusovi why is that so funny lol

bobble

F_spring = -kx

Deusovi

right

so can we calculate that spring constant

bobble

so this will increase because x increase by 5cm

and so F_tens will increase by same amount (to keep at rest)

and this will be the same amount that F_grav for bucket increases, to keep everything nice and at rest

hey wait, that's useful!

Deusovi

yep, looks useful to me!

bobble

03:42

what's the initial mass? 0kg? are we assuming that pencil + bucket weighs nothing?

Deusovi

no idea what the initial mass is -- let's give it a variable

bobble

from problem:

Deusovi

and hope it works out nicely

bobble

> The thread passes
over an ideal pulley and supports a very light bucket. She
attaches a pencil to the bucket; the marking end of the pencil is
in light contact with a piece of paper.

Deusovi

ah, "very light bucket" likely means we should approximate it by ignoring its mass

bobble

03:44

Also, they say mass of cart = M and mass of bucket = m, so let's use those variables

Deusovi

sounds good to me

bobble

F_grav = mg

F_tens = -F_grav = -mg

F_spring = F_tens = -mg

wait, did I mess up some signs?

there, fixed it, right?

Deusovi

you know directions, so let's talk in terms of absolute values

bobble

|F_spring| = |mg| = |kx|

Deusovi

|F_spring| = |F_tens| = |F_grav| = mg

and m/g/k are all positive, so you know mg=kx (where x is amount of displacement to the right)

bobble

03:46

I will say x without any mass = x_0, x with mass is x_f

because they are different

Deusovi

well where are you measuring position from

where is your 0-mark

bobble

oh wait x_0 would just be 0, right?

Deusovi

if x is "displacement from starting position", then yeah

which seems reasonable to me

bobble

so then k = mg/x, and all those are known (m = 0.5kg, g = 10m/s/s, x = 0.05m), but we don't plug in yet

Deusovi

why not?

bobble

03:49

because my teacher says don't plug in? are we allowed to?

would it be okay to plug in now?

Deusovi

it wouldn't produce a false statement, would it?

bobble

no, guess not

Deusovi

it may be more convenient not to plug in and to keep working with variables

but it wouldn't make a false statement if you do

the only potential issue with plugging in numbers early is building up rounding error as you go

bobble

just in case it's not convenient: k = mg/x = 0.5kg * 10m/s/s / 0.05m = 100N/m

Deusovi

hey look the numbers are nice so we don't have to worry about rounding error

almost like this is a problem designed for physics homework or something

3

matt

03:51

oh wait, it is

bobble

can we use T = 2pi * sqrt(m/k) now? or is it still invalid?

Deusovi

when is that equation valid?

bobble

when we are in SHM

which, I guess we can decide to be in now?

Deusovi

not just that

bobble

when no external forces? but that's also required by SHM, I think

Deusovi

03:53

depends on what you mean by "external forces"

bobble

I'm still confused by SHM in general, not entirely sure how I got so far in the homework

Deusovi

(to clarify: i'm thinking through this as we speak. i'm not convinced we can just directly apply that formula here, though it's possible it will still work)

bobble

I think SHM is valid if force is proportional to position, which it still would be, since the bucket pull is constant

I think

Deusovi

SHM just means you're in a situation where the restoring force is proportional to the distance from a certain position

and that produces a sinusoid, because the solution to the differential equation " f′′(x) = k·f(x) " gives you a sine wave

(which you may or may not be familiar with, but hopefully you vaguely understand what i'm saying)

Ankoganit

(for negative k)

Deusovi

03:58

(yes)

bobble

the made-up scenario says this:

> Dominique’s precalculus teacher asks her to make an exact
graph of y = sin x using a real-life process.

Deusovi

(well something something complex numbers. i think.)

bobble

So I assumed that the cart would be in SHM because that would make a sine graph

Deusovi

makes sense

alright yeah pretty sure it will work

because without the mass in the bucket, force is proportional to the distance

if we add a constant force, that'll add a constant distance, but the result will still be proportional

bobble

okay, so T = 2pi * sqrt(m/k). What is m?

Deusovi

04:03

m is the mass that the spring is moving

how much mass is the spring moving?

bobble

so M + m

Deusovi

sounds good to me

bobble

T = 2pi * sqrt((M + m)/k)

T / 2pi = sqrt((M + m)/k)

T^2 / 4pi^2 = (M + m)/k

k * T^2 / 4pi^2 = M + m

oh wait why did I do that?

silly bobble

T = 2pi * sqrt((9.5kg + 0.5kg) / 100N/m)

T = 2pi * sqrt(10kg / 100N/m)

T = 2pi * sqrt(10s^2)

T = 2pi * sqrt(10) s

Deusovi

progress!

bobble

horizontal speed of paper = v, according to problem

v = 0.08m / T

v = 0.08m / (2pi * sqrt(10))s

v = (0.004 * sqrt(10) / pi) m/s

wait, is that it?

Deusovi

04:09

seems good to me

let's double-check -- did you use all the information? if not, is there a reason you didn't?

bobble

yay, thanks!

I'm pretty sure I used everything

mass of cart, mass of bucket

original height of pencil

new middle height of pencil

Deusovi

the only thing you used the heights for was finding out the center position of the pencil

so it wouldn't matter if they were, say, 4 and 1 instead of 2 and 3

does that make sense?

bobble

period in centimeters

yes

the fact that it's SHM, the set-up in general

Deusovi

by "does that make sense" i don't just mean "do you understand the words i'm saying" but "does it make sense in the context of the problem, that if you changed those heights the answer wouldn't change"

bobble

yes, that makes sense. all we cared about was how much the cart moved out, which was determined by how much the pencil moved down

Deusovi

04:14

right, the amplitude shouldn't be a factor in it - shm's period doesn't care about amplitude

so yeah that seems to be everything, looks good to me

bobble

Thank you so much, Deusovi!

Deusovi

no problem!

Prince North Læraðr

06:09

@bobble Okay lol

bobble

21:22

@bobble you made a math error here :) Seriously, me, 10/100 does NOT equal 10

bobble

21:36

@Deusovi, are you available now? I have quick further question on the same physics problem. Sorry for poking :)

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