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

[Asem]

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

I don't understand the question. how am I supposed to find the magnitudes and directions of the velocity from the figure?

 

[berkeman]

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...

 

[Asem]

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.

 

[berkeman]

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?

 

[Asem]

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.

 

[kuruman]

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.

 

[Asem]

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?

 

[kuruman]

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.

 

[jbriggs444]

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.

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:

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.

 

[haruspex]

he is at rest

He? Not if judging from the diagram.

 

[berkeman]

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.

(LOL -- "some Googled up site says"...)

 

[kuruman]

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.