Introduction
In physics, when an object is whirled in a horizontal circle, it requires a centripetal force to keep it moving along its curved path. In this scenario, that force is provided by the tension in the string.
The Physics Concept
For an object of mass $m$ moving at speed $v$ in a circle of radius $r$, the centripetal force $F_c$ is given by: $$F_c = \frac{mv^2}{r}$$
When the stone is whirled horizontally, the tension $T$ in the string must be equal to this centripetal force. The string will break when the tension exceeds the breaking point. Therefore, to find the maximum speed, we set the tension equal to the maximum force the string can sustain.
Step-by-Step Solution
Given:
- Mass ($m$) = $0.8 \text{ kg}$
- Radius ($r$) = $0.9 \text{ m}$
- Maximum Tension ($T_{max}$) = $600 \text{ N}$
Step 1: Set up the equation. $$T = \frac{mv^2}{r}$$
Step 2: Rearrange to solve for velocity $v$. $$v^2 = \frac{T \times r}{m}$$ $$v = \sqrt{\frac{T \times r}{m}}$$
Step 3: Substitute the known values. $$v = \sqrt{\frac{600 \times 0.9}{0.8}}$$ $$v = \sqrt{\frac{540}{0.8}}$$ $$v = \sqrt{675}$$ $$v \approx 25.98 \text{ m/s}$$
Conclusion
The maximum speed the stone can attain without breaking the string is approximately $25.98 \text{ m/s}$.