Angular Velocity of a Large Pendulum on Earth as seen from the stars

In summary: The intent is to ask the student to find the precession rate of the pendulum as a function of latitude.
  • #1
Asem
10
1
Homework Statement
Consider a large simple pendulum that is located at a latitude of 55 degrees N and is swinging in a north-south direction with points A and B being the northernmost and southernmost points of the swing, respectively. A stationary (with respect to the fixed stars) observer is looking directly down on the pendulum at the moment shown in the figure. The Earth is rotating once every 23 h and 56 min. What are the directions (in terms of N, E, W, and S) and the magnitudes of the velocities of the Earth at points A and B as seen by the observer?
Relevant Equations
any
Screenshot 2023-05-07 223118.png


I don't understand the question. how am I supposed to find the magnitudes and directions of the velocity from the figure?
 
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  • #2
Asem said:
What are the directions (in terms of N, E, W, and S) and the magnitudes of the velocities of the Earth at points A and B as seen by the observer?

I don't understand the question. how am I supposed to find the magnitudes and directions of the velocity from the figure?
Yeah, I don't understand it either. Are you sure you copied the question completely? Asking for the motion of the Earth at A and B seems to get rid of the whole "pendulum motion" thing...
 
  • #3
berkeman said:
Yeah, I don't understand it either. Are you sure you copied the question completely? Asking for the motion of the Earth at A and B seems to get rid of the whole "pendulum motion" thing...
it's asking for the velocities of the points corresponding to points A and B on the surface of the Earth not pendulum motion.
 
  • #4
Asem said:
it's asking for the velocities of the points corresponding to points A and B on the surface of the Earth not pendulum motion.
Then why was the pendulum introduced at all in the question? Do they ask a question about the pendulum motion in a follow-up problem maybe?
 
  • #5
berkeman said:
Then why was the pendulum introduced at all in the question? Do they ask a question about the pendulum motion in a follow-up problem maybe?
No, there is no mentioning of a pendulum in this chapter except in this question.
 
  • #6
Asem said:
it's asking for the velocities of the points corresponding to points A and B on the surface of the Earth not pendulum motion.
That's how I interpret the question. Presumably we are looking for the velocities of A and B relative to the observer who is stationary "relative to the fixed stars". The problem is that the Earth, other than its spin about its axis, is also orbiting the Sun while the observer is not. So the orbital velocity of the Earth must be added to the spin velocity of points A and B. I don't think we have enough information to do that because the answer depends on the time of the day (at least) that the observation is made.
 
  • #7
kuruman said:
That's how I interpret the question. Presumably we are looking for the velocities of A and B relative to the observer who is stationary "relative to the fixed stars". The problem is that the Earth, other than its spin about its axis, is also orbiting the Sun while the observer is not. So the orbital velocity of the Earth must be added to the spin velocity of points A and B. I don't think we have enough information to do that.
I think the question ignores the fact that the Earth orbits the Sun. Plus, the Earth's radius is 6.37x10^6 m. Is that enough information to get it?
 
  • #8
Asem said:
I think the question ignores the fact that the Earth orbits the Sun. Plus, the Earth's radius is 6.37x10^6 m. Is that enough information to get it?
Yes, in that case I think that there is enough information. How do you think you should proceed? What would the relevant equation be?

"any" is not a relevant equation, in fact it's not even an equation.
 
  • #9
berkeman said:
Yeah, I don't understand it either. Are you sure you copied the question completely? Asking for the motion of the Earth at A and B seems to get rid of the whole "pendulum motion" thing...
The full text of the problem goes on to ask some follow on questions. The intent of the follow on questions appears to be to get at the precession rate for a Foucault pendulum at the given latitude.
some Googled up site said:
Consider a large simple pendulum that is located at a latitude of 55.0∘N55.0∘N and is swinging in a north-south direction with points A and B being the northernmost and the southernmost points of the swing, respectively. A stationary (with respect to the fixed stars) observer is looking directly down on the pendulum at the moment shown in the figure. The Earth is rotating once every 23 h23 h and 56 min56 min. a) What are the directions (in terms of N, E, W, and S) and the magnitudes of the velocities of the surface of the Earth at points A and B as seen by the observer? [...]
The above is a verbatim match for the problem statement in #1 here. The passage continues:
some Googled up site said:
Note: You will need to calculate answers to at least seven significant figures to see a difference. b) What is the angular speed with which the 20.0−m diameter circle under the pendulum appears to rotate? c) What is the period of this rotation? d) What would happen to a pendulum swinging at the Equator?
The intent is that the observer is hovering above the earth and is momentarily directly above the center of the pendulum. But the observer is not rotating along with the surface of the Earth. In this sense, he she is at rest with respect to the fixed stars. He She is at rest in the non-rotating, earth-centered inertial frame.
 
Last edited:
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  • #10
jbriggs444 said:
he is at rest
He? Not if judging from the diagram.
 
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  • #11
jbriggs444 said:
The above is a verbatim match for the problem statement in #1 here. The passage continues:
Thank you! The problem statement finally makes more sense. :smile:

(LOL -- "some Googled up site says"...)
 
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  • #12
jbriggs444 said:
The intent is that the observer is hovering above the earth and is momentarily directly above the center of the pendulum. But the observer is not rotating along with the surface of the Earth. In this sense, he is at rest with respect to the fixed stars. He is at rest in the non-rotating, earth-centered inertial frame.
In other words, the observer is at rest with respect to the center of the earth but the Force keeps her from falling. Why didn't they say so?

She cannot be at rest with respect to the fixed stars unless she orbits the Sun.
 
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1. What is angular velocity?

Angular velocity refers to the rate of change of the angular position of an object with respect to time. In simpler terms, it is the speed at which an object rotates or moves around a fixed point.

2. How is angular velocity of a pendulum calculated?

The angular velocity of a pendulum can be calculated using the formula ω = √(g/L), where g is the acceleration due to gravity and L is the length of the pendulum. This formula assumes that the pendulum is a simple harmonic oscillator and is valid for small angles of oscillation.

3. Why is the angular velocity of a large pendulum on Earth different from that seen from the stars?

The angular velocity of a pendulum on Earth is affected by the rotation of the Earth itself. This is known as the Coriolis effect and it causes the pendulum to appear to rotate in a different direction when viewed from the stars. This effect is negligible for small pendulums but becomes more significant for larger pendulums.

4. How does the length of the pendulum affect its angular velocity?

The length of the pendulum affects its angular velocity because the formula for calculating angular velocity includes the length of the pendulum. As the length increases, the angular velocity decreases, and vice versa. This is because a longer pendulum takes longer to complete one full swing, resulting in a lower angular velocity.

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

Yes, the angular velocity of a pendulum can be changed by altering either its length or the acceleration due to gravity. For example, the angular velocity can be increased by shortening the length of the pendulum or by increasing the acceleration due to gravity. The angular velocity can also be changed by applying external forces, such as pushing or pulling on the pendulum.

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