To simulate the acceleration of large rockets, the astronauts are spun at the end of long rotating beam of radius 9.8 m . What will be angular velocity required for generating centripetal acceleration 8 times the acceleration due to gravity?

Options

A. `2.828 rad//s`
B. `28.28 rad//s`
C. `282.8 rad//s`
D. zero

Explanation

To solve the problem of finding the angular velocity required to generate a centripetal acceleration that is 8 times the acceleration due to gravity, we can follow these steps:

### Step 1: Understand the relationship between centripetal acceleration, angular velocity, and radius.
The formula for centripetal acceleration (\(a_c\)) is given by:
\[ a_c = \omega^2 \cdot r \]
where:
- \(a_c\) is the centripetal acceleration,
- \(\omega\) is the angular velocity in radians per second,
- \(r\) is the radius of the circular path.

### Step 2: Identify the values given in the problem.
From the problem statement:
- The radius \(r = 9.8 \, \text{m}\)
- The required centripetal acceleration \(a_c = 8g\), where \(g\) (acceleration due to gravity) is approximately \(9.8 \, \text{m/s}^2\).

### Step 3: Calculate the required centripetal acceleration.
Substituting the value of \(g\):
\[ a_c = 8 \cdot 9.8 = 78.4 \, \text{m/s}^2 \]

### Step 4: Substitute the values into the centripetal acceleration formula.
We can rearrange the centripetal acceleration formula to solve for \(\omega\):
\[ \omega^2 = \frac{a_c}{r} \]
Substituting the values we have:
\[ \omega^2 = \frac{78.4}{9.8} \]

### Step 5: Simplify the equation.
Calculating the right side:
\[ \omega^2 = 8 \]

### Step 6: Solve for \(\omega\).
Taking the square root of both sides:
\[ \omega = \sqrt{8} = 2.828 \, \text{rad/s} \]

### Step 7: Round the answer.
Rounding to two decimal places, we get:
\[ \omega \approx 2.83 \, \text{rad/s} \]

### Final Answer:
The angular velocity required to generate a centripetal acceleration 8 times the acceleration due to gravity is approximately \(2.83 \, \text{rad/s}\).

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