As their booster rockets separate, Space Shuttle astronauts typically feel accelerations up to 3g, where g = 9.80 m/s2. In their training, astronauts ride in a device where they experience such an acceleration as a centripetal acceleration. Specifically, the astronaut is fastened securely at the end of a mechanical arm that then turns at constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of 3.44g while in circular motion with radius 9.87 m.
Why did the astronaut go to the circus?
Because they wanted to experience some out-of-this-world centripetal fun! But let's calculate the rotation rate they need to achieve that.
We know that the centripetal acceleration (ac) is given by the formula: ac = ω^2 * r, where ω is the angular velocity (rotation rate) and r is the radius.
To find ω, let's start by rearranging the formula:
ω^2 = ac / r
Plugging in the values given:
ω^2 = (3.44 * g) / r
Since we want the rotation rate in revolutions per second, we can convert it from angular velocity in radians per second using the relation 1 revolution = 2π radians:
ω = (2π * n) / T
Where n is the number of revolutions and T is the time taken in seconds for n revolutions.
Now we can substitute ω into the previous equation:
((2π * n) / T)^2 = (3.44 * g) / r
Let's isolate the variables a bit:
(T^2 * (2π * n)^2) = (3.44 * g * r)
We want the rotation rate in revolutions per second, so we'll rearrange the equation one more time:
n = sqrt((3.44 * g * r) / (T^2 * (2π)^2))
Now we can plug in the given values. But hey, isn't this a lot of math for an astronaut? Maybe they would rather enjoy the view of Earth from space and crack a joke or two with their fellow astronauts.
To determine the rotation rate required to give the astronaut a centripetal acceleration of 3.44g while in circular motion, we can use the following formula:
centripetal acceleration (a) = ω²r
Where:
a = centripetal acceleration
ω = angular velocity
r = radius
Given:
a = 3.44g = 3.44 * 9.80 m/s² (converting g to m/s²)
r = 9.87 m
Substituting the given values into the formula, we have:
3.44 * 9.80 m/s² = ω² * 9.87 m
Solving for ω²:
31.8232 = ω² * 9.87
Dividing both sides of the equation by 9.87:
ω² = 31.8232 / 9.87
ω² ≈ 3.2259
Taking the square root of both sides:
ω ≈ √3.2259
ω ≈ 1.8
The angular velocity ω is equal to the rotation rate in radians per second. To convert it to revolutions per second, we divide by 2π:
Rotation rate = ω / (2π)
Rotation rate ≈ 1.8 / (2π)
Rotation rate ≈ 0.286 rev/s (rounded to three decimal places)
Therefore, the rotation rate required to give the astronaut a centripetal acceleration of 3.44g while in circular motion with a radius of 9.87 m is approximately 0.286 revolutions per second.
To determine the rotation rate (in revolutions per second) required to give an astronaut a centripetal acceleration of 3.44g, we can use the following steps:
Step 1: Convert the centripetal acceleration from g's to meters per second squared (m/s²).
- Given that g = 9.80 m/s² and the centripetal acceleration is 3.44g,
- Multiply the centripetal acceleration (3.44g) by the acceleration due to gravity (9.80 m/s²) to get:
centripetal acceleration = 3.44 * 9.80 m/s² = 33.712 m/s²
Step 2: Use the formula for centripetal acceleration, which relates centripetal acceleration (a), rotation rate (ω), and radius (r):
- a = ω² * r
- Rearrange the formula to solve for ω:
ω = sqrt(a / r)
Step 3: Substitute the values into the formula and calculate ω:
- Given that the centripetal acceleration (a) is 33.712 m/s² and the radius (r) is 9.87 m,
- ω = sqrt(33.712 m/s² / 9.87 m) = sqrt(3.4166) = 1.848 rad/s
Step 4: Convert the rotation rate from radians per second to revolutions per second:
- Since 1 revolution is equivalent to 2π radians,
- ω (rev/s) = ω (rad/s) / (2π) = 1.848 rad/s / (2π) = 0.294 rev/s
Therefore, the rotation rate required to give an astronaut a centripetal acceleration of 3.44g while in circular motion with a radius of 9.87 m is approximately 0.294 revolutions per second.
acceleration= w^2 r
3.44*9.8=w^2 * 9.87
solve for w (in radians per second).
divide that by 2PI to get rps