an astronaut rotates at the end of a tesr machine whose arm has a length of 10.0m. if the acceleration she experiences must not exceed 5g (g=10m/s^2? what is the maximum number of revolutions per minute of the arm?
To find the maximum number of revolutions per minute of the arm, we need to determine the maximum angular velocity the astronaut can experience without exceeding the given acceleration limit.
Let's break down the steps to solve the problem:
1. Convert the acceleration limit from g to m/s^2:
Given acceleration limit: 5g (where g = 10 m/s^2)
Acceleration limit in m/s^2 = 5g = 5 * 10 = 50 m/s^2
2. Calculate the maximum centripetal acceleration:
Using the formula for centripetal acceleration: ac = ω^2 * r
Where ac is the centripetal acceleration, ω is the angular velocity, and r is the arm's length.
Rearranging the formula gives: ω = sqrt(ac / r)
Plugging in the values:
r = 10.0 m (length of the arm)
ac = 50 m/s^2 (maximum centripetal acceleration)
ω = sqrt(50 / 10) = sqrt(5)
3. Convert the angular velocity from radians/second to revolutions/minute:
Since we are looking for the maximum number of revolutions per minute, we need to convert the angular velocity from radians/second to revolutions/minute.
1 revolution = 2π radians
1 minute = 60 seconds
We can use these conversion factors to convert ω to revolutions/minute:
ω (in revolutions/minute) = ω (in radians/second) * (60 seconds / 2π radians)
Plugging in the value of ω = sqrt(5), we get:
ω (in revolutions/minute) = sqrt(5) * (60 / 2π)
4. Calculate the maximum number of revolutions per minute:
Now, we can compute the value of ω (in revolutions/minute) using a calculator:
ω (in revolutions/minute) ≈ 8.5363
Therefore, the maximum number of revolutions per minute of the arm is approximately 8.5363 or rounded to 8.5 revolutions per minute.