how to find the distance traveled by an object moving in sinosoidal motion?

[magomago]

i have a functionn x(t)=Asin(wt) where w=squiggly w and t=time and a=amplitutde

and i found the max velocity it'll have and when that will occur, the max acceleration and when that'll occur, but i can't figure out how to find the distance covered (not the displacement)

does anyone know? I've been staring at the ansewr and all it goes "...since t=2/3 seconds and amplitude=2 then the distance covered in T is 8 cm. so 8cm + 4 cm = 12 cm" and i'm like "huh?"

I know it isn't difficult...but I don't remember i'm missing~

my amplitude is 2cm and my frequency is 1.5Hz~ if anyone could help me out i'd appreciate it~

 

[raptor13]

Arc length can be calculated as follows:

Take the definite integral from A to B of sqrt(f'(x)^2 + 1). It's as simple as that. f'(x), or f'(t) in your case, is just Awcos(t). Square that and you have ((Aw)^2)*cos(t)^2. Add one. Integrate over your range of t (which is 0 to 1, it sounds like) and you've got your answer. Good luck, though, because that integral will be a pain without a calculator or Matlab or something. There is no elementary anti-derivative that I know of.

Hopefully that helps.

 

[magomago]

hrmm i'm supposed to be able to do this with just a scientific calculator (and evne then our professor prefers us to learn how to round~ ie use 10.0m/ses for g if we are just doing rough estimations, etc.)

 

[MetalMat]

Just use your tables in the back of your calculus book, then substitute the constants into the problem. You already know your time and amplitude, should be a cake walk with a scientific calculator.

 

[memo]

by the way squiglly w is probably omega i haven't seen this stuff since sophmore year though so i probably wouldn't be much help