# I need help with this Uniform circular motion problem

## Homework Statement

An astronaut is standing in an space station that spins. The linear speed and the centripedal aceleration that he experiences are bigger on his feet than on his head.

Scientific experiments have proved that a difference of (1/100 )g wont produce this inconveniet for the astronaut.

What should be the radius that the space station must have in comparison with the hight h of the astronaut so that the difference of the aceleration between his head and his feet would be just (1/100) g ?

## The Attempt at a Solution

I dont know how to start to solve this problem can someone please provide help I must solve this problem for tomorrow.

First off, what equations might be important to this problem?

First off, what equations might be important to this problem?

the equations of the uniform rigid motion I guess the equations for the centripedal aceleration v square / radius or (omega square )(radius).
I guess also since it is proportional to the radius the linear velocity equations (radius) (omega) or (2 pi radius) / T.

Also the equations of rigid circular motion that are derivated from cinematics. How can I start to find teh radius?

Well first of all, the question isn't asking for a number value for r, it's just asking for a relationship that has h, the height of the astronaut, in it.

It might help to draw a picture and write out the relationships, I've drawn one to get you started.

The key relationship you want to focus on is:

$$a_{c} = \frac{v^{2}}{r}$$

What's the relationship between a at r and a at r-h??

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gneill
Mentor
The problem statement doesn't mention any preferred spin rate (or angular velocity). The acceleration experienced at a given radius depends upon both the radius and the angular velocity (or the tangential velocity if you wish -- ω2r versus v2/r -- both yield the centripetal acceleration).

With an assumed height for the astronaut (say 6ft, or 1.83m) you can find a relationship between the required spin rate (angular velocity) and the radius. You won't find a specific radius that covers all contingencies.

Well first of all, the question isn't asking for a number value for r, it's just asking for a relationship that has h, the height of the astronaut, in it.

It might help to draw a picture and write out the relationships, I've drawn one to get you started.

The key relationship you want to focus on is:

$$a_{c} = \frac{v^{2}}{r}$$

What's the relationship between a at r and a at r-h??

Im sorry for the delay replying, thanks a lot for the drawing. I guess now from the centripedal aceleration equation a rad (r-h) = v square now I think i should consdier to find another experssion for the linear velocity perhaps v square = initial v +2a s from the cinematic equation or 2 pi r/ T

am I on the right path?

The problem statement doesn't mention any preferred spin rate (or angular velocity). The acceleration experienced at a given radius depends upon both the radius and the angular velocity (or the tangential velocity if you wish -- ω2r versus v2/r -- both yield the centripetal acceleration).

With an assumed height for the astronaut (say 6ft, or 1.83m) you can find a relationship between the required spin rate (angular velocity) and the radius. You won't find a specific radius that covers all contingencies.

is this the right way of solving it a rad(r-h) =(r omega)square ? at this point finding r becomes difficult and I cant solve for r any advice ?

gneill
Mentor
is this the right way of solving it a rad(r-h) =(r omega)square ? at this point finding r becomes difficult and I cant solve for r any advice ?

gneill
Mentor
You only need to consider circular motion equations. The centripetal acceleration is dependent on two parameters: ω and r. ω is the angular velocity at which the space station spins. The linear (tangential) velocity for rotation speed ω at radius r is ωr, if you find that must work with the linear equations; I think though that you'll find the rotational motion equations more direct.

The problem statement is very skimpy on given conditions. For example, they don't specify if there is a desired acceleration value that should be met be at the astronaut's feet. Should it be g to simulate Earth's gravity? Or, is there a required rotation rate?

You can write an equation for the centripetal acceleration at the astronaut's feet and another for the acceleration at his head, and set the difference between them to the desired 100/g. (HINT: centripetal acceleration is ω2r) Then you need to consider your assumptions. Should the acceleration at the periphery of the station be g? Some fraction of g? If so you can specify the relationship between r and ω (use the centripetal acceleration formula again) and replace ω with an appropriate substitution.

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As already stated, the question is terribly worded, is there any more information or is that all that is given to you?

Right now you should have three equations

$$a_{c}(r) = {\omega}_{r}^{2}r$$
and
$$a_{c}(r-h) = {\omega}_{r-h}^{2}(r-h)$$
and
$$a_{c}(r-h) - a_{c}(r) = \frac{g}{100}$$

From there you'll want to solve for r in terms of h and ${\omega}_{i}$