Angular velocity of a pendulum of sorts

In summary, the problem is that when the rod is released from rest, it has a torque which is proportional to cos(theta).
  • #1
El Hombre Invisible
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I feel really dumb asking this... it seems like a really basic problem, but it's due tomorrow, I've got heaps of other stuff to do and I'm not getting anywhere with it.

Here is the problem:

"A uniform rod of length l is freely pivoted at one end. It's initially held horizontal, then released from rest.

First, give angular momentum L as a function of angle."

Now, L is certainly a function of angular velocity: L = Iw. I for a rod is (Ml^2)/3, so L = (Ml^2)w/3. This isn't what I'd have expected for a question asking for L as a function of theta, but a) others I've spoken to claim to have done it this way, and b) I can't figure out a way of getting just theta in there.

"Next, give T (torque) as a function of theta, where theta = 0 when the rod is in its initial position."

Easy enough. Letting x be the distance of an element dm from the pivot point, dT = g cos(theta) x dx, so integrating from x = 0 to x = l gives T = Mgl cos(theta)/2. Also, T = Iw^2.

"Show that w^2 = (3g sin(theta)/l."

Eek! Using the two expressions for T:

T = (Mgl cos(theta))/2 = (Ml^2 w^2)/3,

which I'm presuming is cheating since L is not required here, gives:

w^2 = (3g cos(theta)/2l

which isn't good. I have a cos where I ought to have a sin which comes from my equation for T, but since at theta=0 Mg is perpendicular to the rod, the torque must be highest here so it must be proportional to cos(theta). I also have a 2 I don't want.

I also tried T = dL/dt = (3g cos(theta)/(2l), but I can't figure out a way of writing theta as a function of time to perform the required integral without it swallowing its own tail.

I do have a habit of occassionally going about a simple problem in completely the wrong way at first and I usually come to my senses quite quickly, but I've left the planet with this one. What's annoying me is that the problem should be more complicated than the methods I'm using, but I see no better ones to use.

Any help by tomorrow would be most appreciated.

Thanks,

El Hombre

EDIT: sincerest apologies. In my absence I seem to have forgotten how to use latex. I hope you can read this.
 
Last edited:
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  • #3
Yeah, I cracked it yesterday. Thanks. Also, thanks for not ridiculing my reinvention of the laws of physics (T = Iw^2?? Wha?). I'd had zero sleep the night before I posted and I'd been working on the problem for so long that funny things were happening. After a good nights sleep I got it straight away.
 

Related to Angular velocity of a pendulum of sorts

1. What is angular velocity?

Angular velocity is a measure of how fast an object is rotating around a central point. It is typically measured in radians per second (rad/s).

2. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement (in radians) by the change in time. This can also be represented as the change in angle divided by the change in time.

3. What factors affect the angular velocity of a pendulum?

The angular velocity of a pendulum is affected by its length, mass, and the force of gravity. A longer pendulum with a heavier mass will have a slower angular velocity, while a shorter pendulum with a lighter mass will have a faster angular velocity.

4. How does the angular velocity of a pendulum change over time?

The angular velocity of a pendulum changes over time due to the effects of gravity and air resistance. As the pendulum swings back and forth, it experiences a decrease in angular velocity due to air resistance and an increase in angular velocity due to gravity.

5. Can the angular velocity of a pendulum be constant?

No, the angular velocity of a pendulum cannot be constant. This is because the pendulum is constantly changing direction and experiencing the effects of gravity and air resistance, causing its angular velocity to fluctuate over time.

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