What Happens to a Pendulum's Period at 90 Degrees Inclination?

[racwen]

Homework Statement

What would the period be if the pendulum had been inclined to 90 degrees? What value of g does this correspond to?
theta=90degrees. Just as a side note, theta=0degrees when the arm is facing vertically downwards towards the ground.
g=9.8m/s^2
Effective length=0.275m. The value of length was found from the lab I carried out.
Mass attached to end of arm=0.176kg

Homework Equations

T=2pi x root(L/gcostheta) T is period.
ag=gcostheta. ag is acceleration due to gravity.

The Attempt at a Solution

T=2pi x root(0.275m/(9.8 x cos90))=undefined. Since ag=gcostheta=(9.8 x cos90)=undefined, there would be no acceleration acting on the pendulum, thus no force pushing the pendulum. As a result, the pendulum will not oscillate, and there would be no period. I'm not sure if this is correct, would a pendulum still oscillate when the arm is tilted at 90degrees, and how can period be undefined?

 

[haruspex]

Homework Statement

What would the period be if the pendulum had been inclined to 90 degrees? What value of g does this correspond to?
theta=90degrees. Just as a side note, theta=0degrees when the arm is facing vertically downwards towards the ground.
g=9.8m/s^2
Effective length=0.275m. The value of length was found from the lab I carried out.
Mass attached to end of arm=0.176kg

Homework Equations

T=2pi x root(L/gcostheta) T is period.
ag=gcostheta. ag is acceleration due to gravity.

The Attempt at a Solution

T=2pi x root(0.275m/(9.8 x cos90))=undefined. Since ag=gcostheta=(9.8 x cos90)=undefined, there would be no acceleration acting on the pendulum, thus no force pushing the pendulum. As a result, the pendulum will not oscillate, and there would be no period. I'm not sure if this is correct, would a pendulum still oscillate when the arm is tilted at 90degrees, and how can period be undefined?

What if you think of it in terms of the angle tending to 90 degrees? What does the period tend to?

 

[J Hann]

Remember that the equation T=2pi x root(L/g) T is period was derived
for small angles where sin theta approximately equals theta (usually less than 10 deg).
where did the equation T=2pi x root(L/gcostheta) come from?.

 

[haruspex]

Remember that the equation T=2pi x root(L/g) T is period was derived
for small angles where sin theta approximately equals theta (usually less than 10 deg).
where did the equation T=2pi x root(L/gcostheta) come from?.

racwen is interpreting the problem as a pendulum with a tilted plane of oscillation. Given the question about the value of g, that does seem to be the right view.

 

[J Hann]

OK, so theta in this sense is a constant in the derivation of the equation of a simple equation.

 

[racwen]

What if you think of it in terms of the angle tending to 90 degrees? What does the period tend to?

The period depends on the length, and acceleration (including the angle). I'm not sure if that's what you mean?

 

[haruspex]

The period depends on the length, and acceleration (including the angle). I'm not sure if that's what you mean?

No, I mean suppose it is tilted at some angle theta. You found the period for that. Now, instead of substituting theta equals 90 degrees, consider what happens to the period as theta tends towards 90 degrees. What value does the period tend to?

 

[racwen]

No, I mean suppose it is tilted at some angle theta. You found the period for that. Now, instead of substituting theta equals 90 degrees, consider what happens to the period as theta tends towards 90 degrees. What value does the period tend to?

As period approaches 90 degrees, the period increases. It goes past approximately 25.27s when I used theta=89.9degrees.

 

[haruspex]

As period approaches 90 degrees, the period increases. It goes past approximately 25.27s when I used theta=89.9degrees.

Yes, but what is the value in the limit as theta goes to 90?

 

[racwen]

Yes, but what is the value in the limit as theta goes to 90?

As theta approaches 90, the limit approaches infinity.

 

[haruspex]

As theta approaches 90, the limit approaches infinity.

Right.

 

[racwen]

Right.

So if the period does not exist at 90degrees, does that mean that the pendulum takes an infinite or unknown amount of time to complete one cycle?

 

[haruspex]

So if the period does not exist at 90degrees, does that mean that the pendulum takes an infinite or unknown amount of time to complete one cycle?

The reason I took you through that approach using limits was to demonstrate that it is known.

 

[racwen]

The reason I took you through that approach using limits was to demonstrate that it is known.

Ohh ok, thank you for your help.