A crew of scientists has built a new space station. The space station is shaped like a wheel of radius R with essentially all its mass M at the rim. When the crew arrives the station will be set rotating at a rate that causes an object at the rim to have radial acceleration g thereby simulating Earth's surface gravity. This is accomplished by two small rockets each with thrust T newtons mounted on the station's rim. How long a time t does one need to fire the rockets to achieve the desired condition?

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Solution

Step 1: Given data
Radius of the spaceship in the form of a wheel $= R$
Mass of the spaceship concentrated at its rim $= M$
Radial acceleration of any object placed on the spaceship $= g$
Thrust exerted by the rocket $= T$
Time needed to fire the rockets $= t$
Step 2: Assumptions
Angular velocity $= ω$
Moment of inertia of the spaceship $= I$
Angular acceleration $= α$
Centripetal acceleration $= a_{c}$
Step 3: Calculation of the time needed to fire the rocket
Calculation of the angular velocity
Below is the diagram showing the space station in the form of a rim and the rockets mounted on the station's rim.
The centripetal acceleration must be equal to the acceleration due to gravity so as to simulate the earth's gravity.
Mathematically speaking, $a_{c} = g$ ………….(a)
But the centripetal acceleration of an object is connected to angular velocity and radius by the relation
$a_{c} = ω^{2} R$ …………………………….(b)
Substituting equation (b) in equation (a), we get
$ω^{2} R = g$
$ω = \sqrt{\frac{g}{R}}$ ……………………………(c)
2. Determining the relationship between the angular acceleration, thrust, mass and radius
The torque acting on the spaceship will be equal to the product of the thrust exerted by the rocket and the radius of the rim of the spaceship on which the rockets are mounted.
$𝜏 = T \times R$
Since there are two rockets, therefore total torque will be equal to
$𝜏 = 2 T R$ …………………………….(d)
In rotational mechanics, torque can be expressed as the product of the moment of inertia of a body and its angular acceleration and therefore the torque acting on the spaceship can be expressed as
$𝜏 = I \times α$ ………………………………(e)
Comparing equations (e) and (d), we get
$2 T R = I \times α$
$α = \frac{2 T R}{I}$ ……………………………(f)
Since all the mass of the spaceship is concentrated around its rim, therefore the moment of inertia of the spaceship will be the same as that of the moment of inertia of a point mass.
$I = M \times R^{2}$ …………………….(g)
Substituting equation (g) in equation (f), we get
$α = \frac{2 T R}{M R^{2}}$
$α = \frac{2 T}{M R}$ ……………………..(h)
3. Determining the required relationship by using the derived values of angular acceleration and angular velocity
We know that angular velocity is the product of angular acceleration and time.
$ω = α \times t$ …………………..(i)
Substituting equations (c) and (h) in equation (i), we get
$\sqrt{\frac{g}{R}} = \frac{2 T}{M R} \times t$
$\sqrt{\frac{g}{R}} \times R = \frac{2 T \times t}{M}$
$\sqrt{\frac{g \times R^{2}}{R}} = \frac{2 T \times t}{M}$
$\sqrt{g R} = \frac{2 T \times t}{M}$
$t = \sqrt{g R} \frac{M}{2 T}$
Hence, the required relationship is $t = \sqrt{g R} \frac{M}{2 T}$

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