If a cylindrical space station 300 {\rm m} in diameter is to spin about its central axis, at how many revolutions per minute ({\rm rpm}) must it turn so that the outermost points have an acceleration equal to g?
I do not understand your {\rm m} symbol.
Compute the required angular rotation rate w, in radians per second. Then convert that w rate to rpm. I assume you know how to d that.
The rate is given by
R w^2 = g
w = sqrt(r/R)
To determine the required number of revolutions per minute (rpm) for the cylindrical space station, we need to analyze the acceleration at the outermost points.
First, let's consider the outermost point on the cylindrical space station. When an object rotates in a circular path, the inward force experienced by an object of mass m at this point is provided by the gravitational force acting on it. Therefore, the centrifugal force experienced by the object is equal to its weight:
F_centripetal = F_gravity
To relate this to the acceleration, we can use the following equation:
F_centripetal = m * a_centripetal
F_gravity = m * g
where:
m = mass of the object
a_centripetal = centripetal acceleration
g = acceleration due to gravity
Since the question states that the outermost points of the cylindrical space station should have an acceleration equal to g, we can equate the two equations:
m * a_centripetal = m * g
We can cancel out the mass from both sides:
a_centripetal = g
Now, let's look at the relationship between the centripetal acceleration and the linear speed of an object traveling in a circular path:
a_centripetal = (v^2) / r
where:
v = linear speed
r = radius of the circular path
For a cylindrical space station, the radius is equal to half of the diameter:
r = 300 m / 2 = 150 m
We can rewrite the equation as:
(v^2) / r = g
Simplifying further, we can solve for v:
v^2 = g * r
v = sqrt(g * r)
Now, we need to convert the linear speed to angular speed. The formula for angular speed (ω) is:
ω = v / r
Substituting the value of v, we have:
ω = sqrt(g * r) / r
Simplifying further:
ω = sqrt(g / r)
Finally, we need to convert the angular speed to revolutions per minute (rpm). We know that:
1 revolution = 2π radians
60 seconds = 1 minute
So, to convert from radians per second to revolutions per minute, we can use the following relationship:
1 rpm = (2π radians / 1 revolution) * (1 revolution / 60 seconds) * (60 seconds / 1 minute)
Simplifying, we get:
1 rpm = 2π / 60 radians/second
Therefore, the final formula to calculate the required rpm is:
rpm = (ω * 60) / (2π)
Now, let's plug in the values:
r = 150 m
g = 9.8 m/s^2
Substituting these values into the formula, we get:
ω = sqrt(9.8 / 150)
Finally, calculate the rpm:
rpm = (ω * 60) / (2π) = (sqrt(9.8 / 150) * 60) / (2π)
By evaluating this equation, we can find the specific value for the required rpm of the cylindrical space station.