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We can solve this problem by using the formula for centripetal force, which is related to the tension in the rope:

\[
F = \frac{m v^2}{r}
\]

Where:
- \( F \) is the tension (205 N),
- \( m \) is the mass of the object (0.477 kg),
- \( v \) is the velocity (what we are solving for),
- \( r \) is the radius of the circular path (0.725 m).

Rearranging the formula to solve for velocity \( v \):

\[
v = \sqrt{\frac{F \cdot r}{m}}
\]

Now, substituting the given values:

\[
v = \sqrt{\frac{205 \cdot 0.725}{0.477}} = \sqrt{\frac{148.625}{0.477}} \approx \sqrt{311.5}
\]

\[
v \approx 17.65 \, \text{m/s}
\]

Thus, the maximum speed at which the mass can swing before the rope breaks is approximately **17.65 m/s**.

Would you like more details, or have any questions?

Here are 5 related questions to explore:
1. How would the maximum speed change if the mass was increased?
2. What happens to the tension if the radius of the circle is doubled?
3. How is centripetal force related to the object's velocity?
4. If the rope's breaking tension were higher, what would the new maximum speed be?
5. What is the role of mass in determining the maximum velocity?

**Tip:** Always check units carefully to ensure they are consistent when solving physics problems.

A rope breaks when the tension reaches 205 N. What is the maximum speed at which it can swing a 0.477 kg mass in a circle of radius 0.725 m?

Find the maximum speed at which a 0.477 kg mass can swing in a circle of radius 0.725 m before the rope, which breaks at 205 N, gives way. This physics problem covers the concept of centripetal force and involves using the formula F = mv^2/r. Suitable for high school physics students.

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