How Long to Achieve 1-g Environment in a Rotating Space Cylinder?

[ScottM]

Homework Statement

I'm a Physics I student and having a difficult time with this problem. I feel like I'm missing something obvious but at this point I've put in too much time and I'm ready to ask for help. The problem is, it involves the use of radians as the measure of angular speed and I'm not all that familiar with radians. It's been a long time since I've taken trig (15 years), so I'm guessing this problem is a lot easier than I think. Anyway, here's the problem:

To maintain their bone density and other vital bodily signs, the inhabitants of a cylindrically shaped living space want to generate "1-g environment" on their way to a distant destination. Assume the cylinder has a diameter of 250 m (they live on the inner surface) and is initially not rotating about its long axis. For minimal disruption, the constant angular acceleration is a very gentle .00010 rad/sec-squared. Determine how long it takes for them to reach their goal of a "1-g environment."

I can't see how this done without knowing the mass. I've tried to find an equality with mass on both sides to cancel out, but no luck. I'm having some difficulty with the concepts of angular and tangential speed (specifically the right hand rule, the book does a terrible job of explaining it) as well.

Thanks - Scott

Homework Equations

The Attempt at a Solution

 

[andrevdh]

The spinning cylinder will give its occupants a centripetal acceleration (inwards towards the axis of rotation). The (angular) speed of the drum need to be such that the centripetal acceleration is equal to g (gravitational acceleration)

$$a_c = g = \frac{v^2}{r}$$

The relationship between angular speed of a rotating object and its speed is given by

$$v = \omega r$$

if you substitute this into the equation you will find that the only unknown is the angular speed. You need to find the relation between the angular speed, angular acceleration and time in order to calculate the required time for the cylinder to reach the angular speed that will give the occupants a centripetal acceleration of g.

 

[feryxx]

the angular velocity when the gravitational acceleration is vanished

 

[ashishsinghal]

the angular velocity when the gravitational acceleration is vanished

What do you want to say? I did not understand your post.

 

[rwisz]

Dear Scott,

Thank you for reaching out for help with this problem. I can understand how the use of radians in angular speed and acceleration can be confusing if you haven't used them in a while. However, I can assure you that with some practice and understanding of the concepts, you will be able to solve this problem easily.

Firstly, let's define angular speed and acceleration. Angular speed is the rate at which an object rotates or revolves around a fixed point, measured in radians per second (rad/s). Angular acceleration, on the other hand, is the rate at which the angular speed changes over time, measured in radians per second squared (rad/s^2).

Now, let's look at the problem. The inhabitants of the cylindrical living space want to generate a "1-g environment" by rotating the cylinder about its long axis. This means that the acceleration should be equal to the acceleration due to gravity, which is 9.8 m/s^2. In order to find the time it takes to reach this acceleration, we need to use the formula for angular acceleration, which is α = ω/t, where α is the angular acceleration, ω is the angular speed, and t is the time.

Since the cylinder is initially not rotating, the initial angular speed is 0. Therefore, we can rewrite the formula as α = ω/t = 0.00010 rad/s^2. We also know that the final angular speed is equal to 1 revolution per second, which is equivalent to 2π rad/s. Now, we can solve for t:

α = ω/t
0.00010 rad/s^2 = 2π rad/s / t
t = 2π rad/s / 0.00010 rad/s^2
t = 62831.85 s

Therefore, it will take approximately 62832 seconds for the inhabitants to reach their goal of a "1-g environment."

I hope this explanation helps you understand the concept of angular speed and acceleration better. If you have any further questions, please don't hesitate to ask. Keep up the good work in your Physics I class!

Best regards,

A Scientist