# Arc length

Homework Statement:
An object is undergoing circular motion in horizontal plane at fixed radius## r = 0.12m##
Radial acceleration is ##2+2t ##m/s
Calculate arc length the object swept through the first 2 seconds.
Relevant Equations:
-
From what I understand,

##a_{r} = v_{tan}^2 /r##
##a_{r} = (r\omega)^2 /r##
##a_{r} = r\omega^2##
##\omega^2 = \frac{a_{r}}{r}##
##\omega^2 = \frac{2+2t}{0.12}##
##\omega = \sqrt{\frac{2+2t}{0.12}}##
##s =\int_{0}^{2} \sqrt{\frac{2+2t}{0.12}}##
After integrating, I still can't seem to get the correct answer which is 1.37m
Are my concepts wrong or..?
Thanks

• Delta2

## Answers and Replies

tnich
Homework Helper
##s =\int_{0}^{2} \sqrt{\frac{2+2t}{0.12}}##
What are the units of arc length? What are the units of ##\int \omega dt##?

• Delta2 and jisbon
A
What are the units of arc length? What are the units of ##\int \omega dt##?
Arc length has no units, ##\omega## has a SI units : rad s−1
I realized that when I integrate ##\omega## I will get angular displacement instead of arc length. So after I get the angular displacement I can just multiply it by radius to get the arc length?

EDIT: Solved~! Thanks for the reminder

Last edited:
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
So after I get the angular displacement I can just multiply it by radius to get the arc length?
Or simply go with integrating the tangential velocity from your first expression and never care about angular velocity.

• Delta2
Or simply go with integrating the tangential velocity from your first expression and never care about angular velocity.
Oh yep, that's another alternative :)

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
Arc length has no units
It has dimension length, so the SI unit is metres.

• Orodruin