an astronaut rotates at the end of a test 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?

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To find the maximum number of revolutions per minute (RPM) of the arm, we need to consider the maximum allowable acceleration experienced by the astronaut. Here's how we can find the answer:

1. Start by converting the acceleration limit from g-units to m/s^2. It is given that 1 g = 10 m/s^2, so an acceleration of 5g is equal to 5 * 10 m/s^2, which is 50 m/s^2.

2. Now, let's determine the maximum tangential velocity (v) that the astronaut can experience without exceeding this acceleration limit. The tangential velocity is related to the angular velocity (ω) by the formula v = ω * r, where r represents the length of the arm.

3. Substituting the given values, we have v = ω * r, where r = 10.0 m.

4. Rearranging the equation, we get ω = v / r. Since we have the angular velocity ω in radians per second, we need to convert the tangential velocity v to meters per second.

5. To convert the tangential velocity (v) to meters per second, we need to convert RPM (revolutions per minute) to radians per second. There are 2π radians in one revolution, and one minute is equal to 60 seconds. So, 1 RPM = 2π / 60 radians per second.

6. Substitute the values into the equation ω = v / r and solve for the maximum angular velocity (ω) in radians per second.

ω = (v / r) = (2π / 60) rad/s

7. Now, we need to solve for v using the maximum acceleration.

v = ω * r = ((2π / 60) rad/s) * 10.0 m

8. Finally, we can solve for the maximum number of revolutions per minute (RPM) by rearranging the equation:

RPM = (v * 60) / (2π)

Substitute the value of v from step 7 into the equation to find the maximum number of revolutions per minute.

RPM = (((2π / 60) rad/s) * 10.0 m * 60) / (2π)

Simplify the equation and cancel out the common factors:

RPM = 10

Therefore, the maximum number of revolutions per minute of the arm is 10 RPM.