Simulated Gravity through centrifugal force concept questions

[AFSteph]

Maybe I need to take a little break from my studies but I'm not getting this at all. At the risk of looking really dumb, here's what I need help with;

Imagine that you are living in a rotating space habitat. The habitat has a diameter and a rotational speed such that at the outer edge you experience a simulated gravitational acceleration of 1 g.

A) Imagine that at the outer edge you decide to ride your motorbike in a clockwise direction (the direction of the rotation of the habitat). As your speed increases to 25 km/h does your rotational speed increase, decrease or remain the same? Why? In answering this question, remember (1) that linear speed and tangential speed are the same and (2) tangential speed and rotational speed are proportional.

Solution Attempt: It increases, right? Because tangential/linear speed are proportional?

B) What happens to the simulated gravitational acceleration you experience: does it increase, decrease or remain the same? (In answering this question, consider the relationship between tangential speed and centrifugal force.)

Solution Attempt: I'm not entirely sure what they're hinting at. I guess it increases because centrifugal force is an effect of rotation, so with greater speed the more you feel this effect.

C) If you decide to ride your motorbike in a counter clockwise direction, as your speed increases does your rotational speed increase, decrease or remain the same? Why?

Solution Attempt: It decreases? Because now the speed of the habitat is working against you.

D) What happens to the simulated gravitational acceleration you experience: does it increase, decrease or remain the same?

Solution Attempt: I would assume it decreases because you're moving against the centripetal force, the faster you go, presumably, the more you negate the force.

E) How fast would you have to be going and in what direction to experience weightlessness?

Solution Attempt: I guess in the opposite direction of the environment at the exact same speed, but this is just a guess and I don't really understand why... Nothing like this is covered in my textbook. Sometimes being homeschooled sucks D:

Any added insight would be much appreciated as I'm not really understanding this. Thank you. :)

 

[Simon Bridge]

Imagine that you are living in a rotating space habitat. The habitat has a diameter and a rotational speed such that at the outer edge you experience a simulated gravitational acceleration of 1 g.

It can help with this sort of thing if you think of two different points of view (reference frames).
One is rotating - that's the one where you experience "gravity". The other is fixed - that is the one where you see the rotation.

A) Imagine that at the outer edge you decide to ride your motorbike in a clockwise direction (the direction of the rotation of the habitat). As your speed increases to 25 km/h does your rotational speed increase, decrease or remain the same? Why? In answering this question, remember (1) that linear speed and tangential speed are the same and (2) tangential speed and rotational speed are proportional.

Solution Attempt: It increases, right? Because tangential/linear speed are proportional?

That's the one.
In the fixed frame, your rotational speed has increased because you are driving around the circle faster.
In the rotating frame, you are just travelling.

B) What happens to the simulated gravitational acceleration you experience: does it increase, decrease or remain the same? (In answering this question, consider the relationship between tangential speed and centrifugal force.)

Solution Attempt: I'm not entirely sure what they're hinting at. I guess it increases because centrifugal force is an effect of rotation, so with greater speed the more you feel this effect.

That's good.
In the fixed frame, your "gravity" is supplied because the floor pushes against your feet (the wheels of your bike etc). This is the centripetal force keeping you going in a circle.
When you drive your motorcycle along, the ground has to supply more force to keep you moving in the circle (otherwise you'd drive right through the floor).
In the rotating frame, this is experienced as the floor pushing against your feet (etc) harder... i.e. "gravity" just got stronger.
This is the centrifugal effect.

C) If you decide to ride your motorbike in a counter clockwise direction, as your speed increases does your rotational speed increase, decrease or remain the same? Why?

Solution Attempt: It decreases? Because now the speed of the habitat is working against you.

Kinda - the speed of the habitat neither helps not hinders.
If you were stationary in the rotating frame, then you'd be going at the speed of the habitat in the fixed frame. If you travel against the spin in the rotating frame, you'd just look like you were rotating slower (facing the other way) in the fixed frame.
If you went rally fast against the spin, the fixed frame would see you as stationary - you'd look like a hamster in one of those wheel things. In the rotating frame you'd be going flat out.

D) What happens to the simulated gravitational acceleration you experience: does it increase, decrease or remain the same?

Solution Attempt: I would assume it decreases because you're moving against the centripetal force, the faster you go, presumably, the more you negate the force.

The centripetal force points towards the center - to move against it, you'd have to travel away from the center (which will be "down" from your point of view.

Think in terms of how much force the floor has to supply to keep you going in the circle now you are going slower. Do you know the relationship between centripetal force and tangential velocity?

E) How fast would you have to be going and in what direction to experience weightlessness?

Solution Attempt: I guess in the opposite direction of the environment at the exact same speed, but this is just a guess and I don't really understand why... Nothing like this is covered in my textbook. Sometimes being homeschooled sucks D:

Well I pretty much gave you the answer in a previous comment :)

 

[AFSteph]

Wow! Thanks for helping me again :D

D) What happens to the simulated gravitational acceleration you experience: does it increase, decrease or remain the same?

Solution Attempt: I would assume it decreases because you're moving against the centripetal force, the faster you go, presumably, the more you negate the force.

The centripetal force points towards the center - to move against it, you'd have to travel away from the center (which will be "down" from your point of view.

Think in terms of how much force the floor has to supply to keep you going in the circle now you are going slower. Do you know the relationship between centripetal force and tangential velocity?

Oh duh
Centripetal Force = $\frac{Mass X speed^{2}}{radius of curvature}$

So it will increase as you go faster.

For the last question, I understand that at the halfway point of the outer edge and the center of the habitat, gravity would be 0.5 g. And at the center you would experience 0 g.
That said, I don't think that's the answer they want since it's phrased "how fast and in what direction" Any thoughts?
Thanks again.

 

[Simon Bridge]

Great.
$$F_c=\frac{mv_\perp^2}{r}$$

It's more accurate to say that to go faster in the same radius circle requires more centripetal force.
In this case, the force comes from the floor of the habitat.

I understand that at the halfway point of the outer edge and the center of the habitat, gravity would be 0.5 g. And at the center you would experience 0 g.

That may be a misunderstanding.

It is zero g always.
The experience of gravity comes from surfaces pressing on you ... so there would have to be a floor at R/2 to get you the equivalent g/2. If you started out floating in the center and gave yourself a little push towards the floor you just keep going at a constant speed until you hit it - you'd see the floor rotate under you. (Compare with actual gravity where you accelerate towards the ground.)

You could carry a little jet-pack to stop yourself before you hit - and just hover there over the floor ... you can watch the floor zip past you. People sitting on the floor will see the floor as stationary and you zipping past opposite the direction of rotation.

How fast do they see you go?


OR you can look at it like this - the gravity effect comes from centripetal force.
At what speed is the centripetal force zero?

 

[AFSteph]

Great.
$$F_c=\frac{mv_\perp^2}{r}$$

It's more accurate to say that to go faster in the same radius circle requires more centripetal force.
In this case, the force comes from the floor of the habitat.

That may be a misunderstanding.

It is zero g always.
The experience of gravity comes from surfaces pressing on you ... so there would have to be a floor at R/2 to get you the equivalent g/2. If you started out floating in the center and gave yourself a little push towards the floor you just keep going at a constant speed until you hit it - you'd see the floor rotate under you. (Compare with actual gravity where you accelerate towards the ground.)

You could carry a little jet-pack to stop yourself before you hit - and just hover there over the floor ... you can watch the floor zip past you. People sitting on the floor will see the floor as stationary and you zipping past opposite the direction of rotation.

How fast do they see you go?


OR you can look at it like this - the gravity effect comes from centripetal force.
At what speed is the centripetal force zero?

Oh my. Well, it's almost midnight here and I've answered every question on my exam except this one. So here's what I'm going to write;

E) How fast would you have to be going and in what direction to experience weightlessness?

Solution Attempt: You would need to move to the center (or axis) to experience weightlessness. You would feel as if you weren't moving at all, but from the perspective of people at the outer edge of the habitat, you would appear to be moving at the same rate as the habitat in the opposite direction.

Is that good enough? Thank you so much for your help. Sorry to bother ya.

 

[Simon Bridge]

That won't work no :(
How would you get your motorcycle to the habitat rotational hub?

You've already answered that to experience less gravity, you have to drive opposite the direction of the rotation.
This reduces the centripetal force needed to keep you are a constant distance from the center of rotation.

If the tangential speed of the habitat floor is v, and the motorcycle speed along the floor is u in the opposite direction, then what is the tangential speed of the motorcycle about the hub?
What is the relation between that and the centrifugal force?
What does u have to be to make the centrifugal force zero?

I think that's as close to doing the problem for you as I can get.
Good luck.