Pendulum Formulas Homework: Newton's law & s'' in terms of theta

[sporus]

Homework Statement

1. write Newton's law in terms of s'' theta and g for a pendulum
2. define s'' in terms of theta''

Homework Equations

The Attempt at a Solution

1.
G = -mgsin theta

and G = m s''


but i can't get all three of the required variables into one formula




2.
s'' = -g sin theta

i can't get to theta'' i thought about doing intergrals but i can't figure it out

 

[rock.freak667]

If the pendulum has length l and makes an arc of angle θ, how is the arc length s, related to both l and θ?

 

[Delta2]

for an angle of 2pi (measured in rads), you have arc length 2pi*l, for an angle of theta it is theta*l. Detailed proof of this might need integrals to calculate the arc length or to prove it geometrically: for very small theta $$d\theta$$ the arc length ds would be equal to the string length dr and equal to 2lsin($$d\theta/2$$). But because $$d\theta$$ is small sin($$d\theta/2$$)=$$d\theta/2$$. So ds=l$$d\theta$$ and thus by simple integration s=l$$\theta$$.

 

[sporus]

i don't get it still ;;

 

[rock.freak667]

i don't get it still ;;

Delta² shows you the derivation to arrive at the relation of s=lθ.

 

[sporus]

does that mean that what he posted above was integration? i didn't get that because we haven't covered that in calc yet. i'll go over it again and try to get it

 

[rock.freak667]

does that mean that what he posted above was integration? i didn't get that because we haven't covered that in calc yet. i'll go over it again and try to get it

It essentially becomes the same as finding the arc length given the radius l and and angle θ.