A large, doughnut-shaped section of a spacecraft is being designed for long-term space flight. The outer wall of the rotating section will become the floor for the astronauts, and the centripetal acceleration produced by the floor will simulate gravity.

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If this section of the spacecraft has a radius of 300.0 m, what speed will produce an acceleration of 9.80 m/s2?

To determine the speed that will produce an acceleration of \( 9.80 \, \text{m/s}^2 \), we need to use the formula for centripetal acceleration:

\[
a_c = \frac{v^2}{r}
\]

where:
- \( a_c \) is the centripetal acceleration,
- \( v \) is the linear speed,
- \( r \) is the radius of the circular path.

### Given:
- \( a_c = 9.80 \, \text{m/s}^2 \) (acceleration needed to simulate gravity),
- \( r = 300.0 \, \text{m} \) (radius of the rotating section).

Now, solve for \( v \):

\[
v = \sqrt{a_c \cdot r}
\]

Substitute the given values:

\[
v = \sqrt{9.80 \, \text{m/s}^2 \times 300.0 \, \text{m}}
\]

Now, calculate the speed \( v \):The speed required to produce an acceleration of \( 9.80 \, \text{m/s}^2 \) is approximately \( 54.22 \, \text{m/s} \).

Would you like further details or explanations on this? 

Here are 5 related questions to extend this discussion:
1. How is centripetal acceleration related to rotational velocity?
2. What would happen if the radius were doubled? How would it affect the required speed?
3. How does the mass of an object affect centripetal acceleration?
4. Can centripetal force be used to simulate gravity in smaller spacecraft sections?
5. What are some real-world examples of centripetal forces in space travel?

**Tip:** The larger the radius of rotation, the lower the speed required to simulate the same gravitational effect.

A large, doughnut-shaped section of a spacecraft is being designed for long-term space flight. The outer wall of the rotating section will become the floor for the astronauts, and the centripetal acceleration produced by the floor will simulate gravity. If this section of the spacecraft has a radius of 300.0 m, what speed will produce an acceleration of 9.80 m/s²?

Learn how to determine the speed required to produce simulated gravity in a rotating spacecraft with a radius of 300.0 m. This problem uses the centripetal acceleration formula a_c = v² / r and involves key physics concepts of circular motion. Suitable for high school students in Grades 11-12.

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