The problem you shared is about circular motion, particularly focusing on an astronaut in a centrifuge. Let's go step by step through the questions:

Given:

• Radius $r = 5.00 \, \text{m}$
• Centripetal acceleration $a_c = 6.90g$, where $g \approx 9.81 \, \text{m/s}^2$.

So, $a_c = 6.90 \times 9.81 = 67.689 \, \text{m/s}^2$.

(a) What is the astronaut's speed?

The centripetal acceleration is related to the speed $v$ by the formula: $a_c = \frac{v^2}{r}$ Solving for $v$: $v = \sqrt{a_c \times r} = \sqrt{67.689 \times 5.00} \approx 18.41 \, \text{m/s}$

(b) How many revolutions per minute are required to produce this acceleration?

First, we need to find the angular velocity $\omega$ in radians per second. The formula linking speed to angular velocity is: $v = \omega r \quad \text{or} \quad \omega = \frac{v}{r}$ Substituting $v = 18.41 \, \text{m/s}$ and $r = 5.00 \, \text{m}$: $\omega = \frac{18.41}{5.00} \approx 3.682 \, \text{rad/s}$ Now, to find the revolutions per second (rps), we convert from radians per second to revolutions per second using the fact that $2\pi \, \text{radians} = 1 \, \text{revolution}$: $\text{rps} = \frac{\omega}{2\pi} = \frac{3.682}{2\pi} \approx 0.586 \, \text{rps}$ Finally, to find revolutions per minute (rpm): $\text{rpm} = 0.586 \times 60 \approx 35.17 \, \text{rpm}$

(c) What is the period of the motion?

The period $T$ is the time it takes to complete one full revolution. It is related to the frequency by: $T = \frac{1}{f}$ where $f$ is the frequency in revolutions per second. From part (b), we found $f = 0.586 \, \text{rps}$, so: $T = \frac{1}{0.586} \approx 1.71 \, \text{seconds}$

Let me know if you'd like any more details on the solution, or if you have any questions!

Follow-up questions:

1. How does changing the radius affect the speed required to maintain the same centripetal acceleration?
2. What is the relationship between angular velocity and centripetal acceleration?
3. How would the number of revolutions per minute change if the astronaut's centripetal acceleration was doubled?
4. What would happen to the period if the radius was halved?
5. How is centripetal force related to mass in this scenario?

Tip:

In circular motion, increasing the radius while keeping centripetal acceleration constant will reduce the speed needed to maintain that acceleration.