Directions of centripetal acceration and gravity on Earth?

[Elvis 123456789]

Im picturing a stationary object at the equator of the earth, going around in uniform circular motion due to the Earth's rotation. The centripetal acceleration would point directly inwards into the Earth at all points in the circular rotation, and gravity would also point in the same direction.

This is how I am visualizing it. I have read that the accelerations oppose each other, but I cannot picture why.

 

[Suraj M]

This is how I am visualizing it. I have read that the accelerations oppose each other, but I cannot picture why.

About which acceleration did you read?
Centripetal? Are you sure?

 

[Elvis 123456789]

About which acceleration did you read?
Centripetal? Are you sure?

The main question was this:
The Earth has a radius of 6380 km and turns around once on its axis in 24 h.

a) What is the radial acceleration of an object at the Earth's equator? Give your answer in m/s^2.

b) What is the radial acceleration of an object at the Earth's equator? Give your answer as a fraction of g.

c) If a_rad at the equator is greater than g, objects would fly off the Earth's surface and into space. What would the period of the Earth's rotation have to be for this to occur?

I have the answers, but the last part "c" implies that the directions of the centripetal acceleration due to Earth's rotation goes against gravity, and I cannot see why.

 

[Suraj M]

Which force is centripetal and which centrifugal, is what you need to identify.

 

[Elvis 123456789]

Which force is centripetal and which centrifugal, is what you need to identify.

I don't see why either of them would be centrifugal. I know that gravity always points toward the earth, but wouldn't the other force point towards the Earth at all points in the rotation as well? I mean obviously not, but I don't see it.

 

[Suraj M]

Have you ever stood on a rotating table? Carousel?
There you are holding on that's the centripetal and you feel something pushing you away from the center, radially, that is the centrifugal force
Try relating that, here.
Which $“other force”$ are you talking about?

 

[Elvis 123456789]

I see what you're saying, but I am still confused. In the problem I posted above, the radial acceleration due to the Earth's rotation is apparently pointing outward, but I don't see why it would. The acceleration is supposed to point into the circle, the circle being the path traced by the Earth's rotation. I drew a picture of how I thought the problem was so maybe you can see the error in my reasoning

pardon my bad shapes

 

[Suraj M]

You aren't taking centrifugal force
For minute think about it practically
The force due to rotation needs to be outward, right?

 

[Elvis 123456789]

You aren't taking centrifugal force
For minute think about it practically
The force due to rotation needs to be outward, right?

Sorry but I guess I have a significant gap in my learning. My physics 1 never talked about any centrifugal forces. I haven't seen anything about centrifugal forces in the solutions concerning this problem either. For instance, this solution doesn't even talk about any forces http://scalettar.physics.ucdavis.edu/p9a/midterm1_p9A_2007_secCKey.pdf

just the two accelerations. And the big problem I am having is seeing why this accelerations oppose each other.

 

[Suraj M]

Check the examples section here:

https://en.m.wikipedia.org/wiki/Centrifugal_force

 

[Elvis 123456789]

Check the examples section here:

https://en.m.wikipedia.org/wiki/Centrifugal_force

This was a great link, thank you very much. I think I understand a bit better now. So to clarify, looking at an object at the Earth's equator, with the Earth being our frame of reference, it is necessary to introduce the "centrifugal force" which opposes gravity in order to account for the contribution of gravity to the centripetal acceleration?

 

[mfig]

I have read that the accelerations oppose each other, but I cannot picture why.

If the Earth were to spin fast enough, we would fly outward just like we do on a merry-go-round. The centripetal force you are thinking of, which points inward, is the force necessary to keep us at our distance from the axis of rotation of the Earth in the absence of gravity.

Picture this: You are in deep space, far away from any mass. What you calculated is the tension force a rope would have if it were R_earth long and you were being spun in a circle at a rate of 24 hours per revolution. But it is not a rope holding you at a distance of R_earth from the Earth's rotational axis, it is gravity that holds you to the Earth's surface. Gravity doesn't care whether we are spinning or sitting still - it pulls us the same either way. But if the rotation rate of the Earth were to gradually accelerate, it would eventually reach a point where we would be flung off even though gravity tries to pull us back. What that tells you is that there is some kind of a force balance going on between:

1. A force that wants to pull us off the earth. This force depends on the rotational speed of the earth.
2. A force that wants to pull us to the earth. This is gravity and does not depend on the rotational speed of the earth.

Of course, the long story is that there is no such force as a centrifugal force in reality. But we act as if there were for convenience in calculations. You can read more about centrifugal forces here.

 

[Suraj M]

Nice video

 

[mfig]

Nice video

Yeah, I can't tell if he blacked out before he was flung off or what. I know people can faint from hi g-forces, and that is what it looks like might have happened to him. I am no expert in such matters, so I could be totally wrong.

 

[ehild]

Sorry but I guess I have a significant gap in my learning. My physics 1 never talked about any centrifugal forces. I haven't seen anything about centrifugal forces in the solutions concerning this problem either. For instance, this solution doesn't even talk about any forces http://scalettar.physics.ucdavis.edu/p9a/midterm1_p9A_2007_secCKey.pdf

just the two accelerations. And the big problem I am having is seeing why this accelerations oppose each other.

You do not need centrifugal force, if you use a rest frame of reference. There is a body on the equator, moving together with the Earth, on circular path. Knowing the radius of the Earth and its angular velocity, you can find the radial (centripetal) acceleration and the centripetal force needed to that circular motion.
The centripetal force is the resultant of the force of gravity and the normal force acting on the object by the surface of the earth.
If the Earth rotated so fast that negative normal force would be needed to keep the object on the circle of radius R, the object would fly away from the Earth surface. The minimum speed is when the normal force needed is zero, so the froce of gravity is equal to the centripetal force.

 

[ehild]

You aren't taking centrifugal force
For minute think about it practically
The force due to rotation needs to be outward, right?

The problem can be solved both from a rest frame of reference or the fame of reference rotating together with the Earth. We have centrifugal force in the rotating frame of reference. The OP did not learned about accelerating frames yet, so he needs help according to his level of knowledge.

 

[Suraj M]

The problem can be solved both from a rest frame of reference or the fame of reference rotating together with the Earth. We have centrifugal force in the rotating frame of reference. The OP did not learned about accelerating frames yet, so he needs help according to his level of knowledge.

I was just trying to point out that the force should be outward. Sorry[emoji17]

 

[Elvis 123456789]

You do not need centrifugal force, if you use a rest frame of reference. There is a body on the equator, moving together with the Earth, on circular path. Knowing the radius of the Earth and its angular velocity, you can find the radial (centripetal) acceleration and the centripetal force needed to that circular motion.
The centripetal force is the resultant of the force of gravity and the normal force acting on the object by the surface of the earth.
If the Earth rotated so fast that negative normal force would be needed to keep the object on the circle of radius R, the object would fly away from the Earth surface. The minimum speed is when the normal force needed is zero, so the froce of gravity is equal to the centripetal force.
View attachment 94829

Your explanation is the one I defaulted to because it made a lot more sense to me. In terms of forces I can understand the problem. My misunderstanding with this question comes from the fact that I was trying to explain it to someone who hadn't covered Newton's laws yet. So the problem that I posted asks for the period that would correspond to a speed at which an object would fly off the surface of the Earth. The condition given is that the normal force would have to be zero for an object to fly off the surface of the Earth. But how would you explain this to someone who hasn't even covered forces yet? That is, just using the concept of accelerations, and circular motion.

 

[ehild]

Your explanation is the one I defaulted to because it made a lot more sense to me. In terms of forces I can understand the problem. My misunderstanding with this question comes from the fact that I was trying to explain it to someone who hadn't covered Newton's laws yet. So the problem that I posted asks for the period that would correspond to a speed at which an object would fly off the surface of the Earth. The condition given is that the normal force would have to be zero for an object to fly off the surface of the Earth. But how would you explain this to someone who hasn't even covered forces yet?

Tell him you will explain when he learned about forces and Newton's Laws. You can show him the Merry-go-round video to get the feeling what happens, without explanation.

 

[ehild]

I was just trying to point out that the force should be outward. Sorry[emoji17]

Which force?

 

[A.T.]

I have the answers, but the last part "c" implies that the directions of the centripetal acceleration due to Earth's rotation goes against gravity, and I cannot see why.

No, they both point inwards. Gravity provides the centripetal acceleration, but at some spin rate the required centripetal acceleration would be greater than gravity can provide. You have to find that spin rate.