Max velocity for simple pendulum oscillation

In summary, the velocity of the simple pendulum changes with angle and the tangential velocity is found by dividing through by the distance traveled.
  • #1
PhizKid
477
1

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


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


The Attempt at a Solution


The closest resemblance I could find was using energy:

[itex]\frac{1}{2}kx^2 = \frac{1}{2}mv_{max}^2[/itex] 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: [itex]v_{max} = \omega_{frequency} \cdot x[/itex]

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?
 
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  • #2
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.
 
  • #3
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)
 
  • #4
Simon Bridge said:
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?

rock.freak667 said:
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.
 
  • #5
PhizKid said:
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.
 
  • #6
PhizKid said:
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##.
 

FAQ: Max velocity for simple pendulum oscillation

1. What is the maximum velocity of a simple pendulum?

The maximum velocity of a simple pendulum depends on the length of the pendulum and the force of gravity. It can be calculated using the equation v = √(gl), where v is the maximum velocity, g is the acceleration due to gravity, and l is the length of the pendulum. For example, a simple pendulum with a length of 1 meter would have a maximum velocity of approximately 3.13 m/s.

2. How does the maximum velocity of a simple pendulum change with length?

The maximum velocity of a simple pendulum is directly proportional to the square root of its length. This means that as the length of the pendulum increases, the maximum velocity also increases. However, this relationship is not linear, as a doubling of the length will result in a less than doubling of the maximum velocity.

3. Can the maximum velocity of a simple pendulum be greater than the initial velocity?

No, the maximum velocity of a simple pendulum is always less than the initial velocity. This is because the pendulum loses energy due to friction and air resistance, causing it to slow down over time. The initial velocity is the maximum velocity the pendulum will have, and it will decrease from there.

4. How does the mass of the pendulum affect the maximum velocity?

The mass of the pendulum does not affect the maximum velocity. As long as the length and force of gravity remain constant, the maximum velocity will also remain constant. The mass may affect other factors such as the period and amplitude of the pendulum's oscillations, but not the maximum velocity.

5. Is there a limit to the maximum velocity of a simple pendulum?

Technically, there is no limit to the maximum velocity of a simple pendulum. However, as the length of the pendulum approaches infinity, the maximum velocity approaches the speed of light, making it practically impossible to achieve. In most cases, the maximum velocity will be limited by the length and force of gravity, resulting in a finite maximum velocity.

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