A stone is attached to a rope of length l=80 cm is rotated with a speed of 240 rpm. At the moment when the velocity is directed vertically upwards, the rope breaks. To what height does the stone rise further?

Options

A. 1.2 m
B. 41.2 m
C. 20.6 m
D. 24.9m

Answer

To solve the problem step by step, we need to determine the height the stone rises after the rope breaks. Here’s how we can approach it:

### Step 1: Convert RPM to Angular Velocity
The stone is rotating at a speed of 240 revolutions per minute (RPM). We need to convert this to angular velocity (ω) in radians per second.

\[ \omega = 2\pi \times \frac{n}{60} \]

Where \( n = 240 \) RPM.

\[ \omega = 2\pi \times \frac{240}{60} = 8\pi \, \text{rad/s} \]

### Step 2: Calculate the Linear Velocity
The linear velocity (v) of the stone can be calculated using the formula:

\[ v = r \cdot \omega \]

Where \( r \) is the length of the rope, which is 0.8 m (80 cm).

\[ v = 0.8 \cdot 8\pi = 6.4\pi \, \text{m/s} \]

### Step 3: Apply Conservation of Energy
When the rope breaks, the stone will rise until its velocity becomes zero. We can use the conservation of mechanical energy principle, where the kinetic energy at the point of breaking equals the potential energy at the maximum height.

The kinetic energy (KE) at point B (when the rope breaks) is:

\[ KE = \frac{1}{2} mv^2 \]

The potential energy (PE) at the maximum height (point E) is:

\[ PE = mgh \]

Setting these equal gives:

\[ \frac{1}{2} mv^2 = mgh \]

### Step 4: Cancel Mass and Rearrange
Since mass (m) appears on both sides, we can cancel it out:

\[ \frac{1}{2} v^2 = gh \]

Rearranging for height (h):

\[ h = \frac{v^2}{2g} \]

### Step 5: Substitute the Values
Now we substitute \( v \) and \( g \) (where \( g \approx 9.8 \, \text{m/s}^2 \)) into the equation:

\[ h = \frac{(6.4\pi)^2}{2 \cdot 9.8} \]

Calculating \( (6.4\pi)^2 \):

\[ (6.4\pi)^2 \approx (20.106)^2 \approx 404.25 \]

Now substituting this value into the height equation:

\[ h = \frac{404.25}{19.6} \approx 20.62 \, \text{m} \]

### Conclusion
Thus, the stone rises to a height of approximately **20.62 meters** after the rope breaks.

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