Air-track cart oscillation: finding time when x=4.6 cm

[jaysea]

please help, homework trouble, springs and ocilatory motion!

Homework Statement

The position of a an air-track cart that is oscillating on a spring is given by (16.5cm)cos[(15.0s-1)t].
At what value t after t=0 is the cart first located at x=4.6 cm?

Homework Equations

x=Acos[(angular freqency)(t)+(phase constant)] i think is the only one but i don't get the question right, maybe its my math.

The Attempt at a Solution

@=phase constant, w= angular frequency

comparing x= 4.6cm=(16.5cm)cos[(15.0s-1)t] to:
X=Acos[wt+@]
inserting the information we get
0.046m=0.165m*cos1.5(t)
0.278=cos1.5t
t=0.278s

but incorrect.

 

[Mentz114]

I don't see a phase constant in this -

(16.5cm)cos[(15.0s-1)t]

which has no constant in the [..] because it's all multiplied by t. Also the dimensions are wrong. Are you sure you haven't miscopied it ?

 

[jaysea]

the question itself is copied n pasted. the reason there's no phase constant is because apparently at that point phase constant = 0.. i don't know maybe I am wrong about that but.. but the question is copied n pasted.

 

[Mentz114]

Let me explain why I think it's wrong.

Anything inside a cos() function must be dimensionless, like an angle. It cannot be in meters or Kg etc.
Therefore the expression you've got

15s-1 must be in units of 1/t so that when multiplied by t it becomes an angle. But 1 times t is in seconds and cannot be an angle.

Usually its cos(wt+phi) as you've written, where w is 1/t and phi is already an angle.

 

[jaysea]

im confuzed right now :s

 

[Mentz114]

OK, I've figured out your weird notation The 15s-1 is 15 radians per second. So your problem is straightforward. w is 15 and phase constant is zero.

So use 15 in your calculation, not 1.5.

 

[jaysea]

ugh it still didnt work out right, i got like 0.0289 >_<

 

[jaysea]

then i got 0.289 when i tried again, but it still wasnt right

 

[Mentz114]

I get

t = [arccos( 0.278)]/15

but I don't have arccos tables ...

 

[jaysea]

if i do that then i get 4.92 which apparently still aint right, unless I am doing something wrong mathwise, i took the inverse cosine of 0.278 which was 73.9then divided it by 15. and got 4.92.

 

[Mentz114]

The angle must be in radians, so 73.9 deg = 73.9/57.295 rad = 1.289 rad

You'll get about .085 ...

( 57.295 = 180/pi )

a complete cycle only takes about 0.42 secs ( 2*pi/15)

 

[jaysea]

WOO who i got it! thanks!

 

[Mentz114]

Well done. Sorry I misunderstood the 15s-1.