A stone is tied to one end of a string. On holding the other end, the string is whirled in a horizontal plane with progressively increasing speed. It breaks at some speed because

Options

A. the grvitational forces of the Earth is greater than the tension in string.
B. the required centripetal force is greater than the tension sustained by the string.
C. the required centripetal force is less than the tension in the string.
D. the centripetal force is greatr than the weight of the stone.

Answer

To solve the problem, we need to understand the dynamics of circular motion and the forces involved when a stone is whirled in a horizontal plane by a string. Here’s a step-by-step breakdown of the reasoning:

### Step 1: Understand the Setup
A stone is tied to one end of a string, and the other end is held while the stone is whirled in a horizontal plane. As the speed of the stone increases, it experiences circular motion.

**Hint:** Visualize the scenario by drawing a circle with the stone at the edge and the center where the string is held.

### Step 2: Identify the Forces Acting on the Stone
In circular motion, the stone experiences a centripetal force that keeps it moving in a circle. This force is provided by the tension in the string. The gravitational force acts downward but does not affect the horizontal motion directly.

**Hint:** Remember that the tension in the string is what provides the necessary centripetal force for circular motion.

### Step 3: Write the Equation for Centripetal Force
The centripetal force \( F_c \) required to keep the stone moving in a circle is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the stone,
- \( v \) is the speed of the stone,
- \( r \) is the radius of the circular path.

**Hint:** Note that as the speed \( v \) increases, the required centripetal force \( F_c \) also increases.

### Step 4: Relate Tension and Centripetal Force
In normal conditions, the tension \( T \) in the string provides the centripetal force:
\[ T = \frac{mv^2}{r} \]

**Hint:** Understand that the string can only provide a certain maximum tension before it breaks.

### Step 5: Determine the Breaking Point of the String
Every material has a limit to the tension it can withstand, known as the breaking strength. If the required centripetal force exceeds this breaking strength, the string will break.

**Hint:** Think about what happens when you pull a rubber band too far; it snaps when the tension exceeds its limit.

### Step 6: Conclusion
As the speed of the stone increases, there comes a point where the required centripetal force (which is proportional to the square of the speed) becomes greater than the maximum tension the string can handle. At this point, the string breaks.

**Final Statement:** The string breaks because the required centripetal force becomes greater than the tension that the string can sustain.

### Final Answer
The correct reasoning is that the string breaks because the required centripetal force is greater than the tension sustained by the string.