# I with this Uniform circular motion problem

• Jimmy84
In summary, the problem asks for a relationship between the radius of a spinning space station and the height of an astronaut standing on it. The equations for the uniform rigid motion and the equations for the circular motion are derivated from cinematics. First, to find the radius required, the problem asks for a relationship between the height and the angular velocity or spin rate of the space station. Second, to find the radius, you need to focus on the relationship between the height and the linear velocity. The problem doesn't specify which angular velocity or spin rate should be used, so you will need to use whatever equation is applicable for the situation. Finally, to find the radius, you need to use the relationship between
Jimmy84

## 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 won't 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 don't 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?

Clever-Name said:
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??

#### Attachments

• fdsa.jpg
9 KB · Views: 521
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.

Clever-Name said:
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?

gneill said:
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 can't solve for r any advice ?

Jimmy84 said:
is this the right way of solving it a rad(r-h) =(r omega)square ? at this point finding r becomes difficult and I can't solve for r any advice ?

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.

Last edited:
gneill said:

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 ${\omega}_{i}$

## 1. What is uniform circular motion?

Uniform circular motion is a type of motion in which an object moves in a circular path at a constant speed. This means that the object covers the same distance in the same amount of time.

## 2. How is uniform circular motion different from linear motion?

In uniform circular motion, the object moves in a circular path, while in linear motion, the object moves in a straight line. Additionally, in uniform circular motion, the speed of the object remains constant, while in linear motion, the speed can vary.

## 3. What is the role of centripetal force in uniform circular motion?

Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle and is necessary to maintain the object's constant speed. Without centripetal force, the object would move in a straight line tangent to the circle.

## 4. How is the period of uniform circular motion related to the radius of the circle?

The period of uniform circular motion is directly proportional to the radius of the circle. This means that as the radius increases, the period also increases, and vice versa. This relationship can be represented by the equation T = 2πr/v, where T is the period, r is the radius, and v is the speed of the object.

## 5. How can I calculate the speed of an object in uniform circular motion?

The speed of an object in uniform circular motion can be calculated using the equation v = 2πr/T, where v is the speed, r is the radius, and T is the period. This equation is derived from the relationship between speed, distance, and time, and is also known as the tangential speed formula.

• Introductory Physics Homework Help
Replies
12
Views
2K
• Introductory Physics Homework Help
Replies
10
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
2K
• Introductory Physics Homework Help
Replies
38
Views
3K
• Introductory Physics Homework Help
Replies
1
Views
1K
• Mechanics
Replies
23
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
23
Views
471