Max velocity for simple pendulum oscillation

[PhizKid]

Homework Statement

Show that v_max = w_max * Length of string where v_max is the velocity of the simple pendulum and w_max is the maximum angular velocity.

Homework Equations

\omega_{velocity} = -\theta_{max} \cdot \omega_{frequency} \cdot sin(\omega_{frequency} \cdot t + \phi)

The Attempt at a Solution

The closest resemblance I could find was using energy:

\frac{1}{2}kx^2 = \frac{1}{2}mv_{max}^2 where x = some displacement. The displacement in this pendulum's clase would be the Length of the string * angular displacement because s = Lθ.

Solving for v_max gives: v_{max} = \omega_{frequency} \cdot x

But the 'x' in this case represents the displacement made by the pendulum (the arclength subtended by the angular displacement).

Is there another equation I'm unaware of?

 

[Simon Bridge]

Well yes - via $v_{T}=R\omega$ perhaps?

I think, to understand the problem, you need to say where you have to start from
... possibly by deriving the equation for "v" - however it has been defined.
I don't think you have been very careful with the definitions of terms.

 

[rock.freak667]

You may need to solve the SHM DE of

d²x/dt² + ω²x = 0

Are you able to solve this DE or know the solutions to this DE? (it is similar to your relevant equation)

 

[PhizKid]

Well yes - via $v_{T}=R\omega$ perhaps?

I think, to understand the problem, you need to say where you have to start from
... possibly by deriving the equation for "v" - however it has been defined.
I don't think you have been very careful with the definitions of terms.

Oh, so there is an equation that states $v_{T}=R\omega$. I did not know this. How do you derive this?

You may need to solve the SHM DE of

d²x/dt² + ω²x = 0

Are you able to solve this DE or know the solutions to this DE? (it is similar to your relevant equation)

My relevant equation is just the derivative to this solution. I didn't think the angular position equation was more useful than its derivative so I only included the equation for angular velocity.

 

[rock.freak667]

My relevant equation is just the derivative to this solution. I didn't think the angular position equation was more useful than its derivative so I only included the equation for angular velocity.

Yes and that is why I thought by show, you would have had to show the steps in the derivation towards the DE.

 

[Simon Bridge]

Oh, so there is an equation that states $v_{T}=R\omega$. I did not know this. How do you derive this?

<puzzled> the pendulum is an example of circular motion:

The distance $s$ on an arc, radius $r$, changes with angle $\theta$ as $ds=r d\theta$ (follows from the definition of "angle"). To get the tangential velocity $v_{T}$ just divide through by $dt$.