A particle of mass `m` attached to a string of length `l` is descending circular motion on a smooth plane inclined at an angle `alpha` with the horizontal. For the particle to reach the highest point its velocity at the lowest point should exceed.

Options

A. `sqrt(5gl)`
B. `sqrt(5gl(cos alpha+1))`
C. `sqrt(5gl//tan alpha)`
D. `sqrt(5gl sin alpha)`

Answer

To solve the problem, we need to determine the minimum velocity required at the lowest point of the circular path for the particle to reach the highest point in its motion. Here’s a step-by-step solution:

### Step 1: Understand the Forces Acting on the Particle
When the particle is at the lowest point of its circular motion, it experiences two forces:
- The gravitational force acting downward, which can be resolved into two components: one along the incline \( mg \sin \alpha \) and one perpendicular to the incline \( mg \cos \alpha \).
- The tension in the string acting upward.

### Step 2: Determine the Effective Gravitational Force
At the lowest point, the effective gravitational force acting on the particle can be considered as \( g \sin \alpha \). This is because the particle is moving along the inclined plane, and we only need to consider the component of gravitational force acting down the incline.

### Step 3: Apply the Condition for Circular Motion
For the particle to complete the circular motion and reach the highest point, it must have sufficient velocity at the lowest point. The centripetal force required to keep the particle in circular motion at the lowest point is provided by the tension in the string and the gravitational force acting on the particle.

### Step 4: Use the Formula for Velocity at the Lowest Point
The minimum speed \( v \) at the lowest point for the particle to reach the highest point in a vertical circular motion is given by the formula:
\[ v = \sqrt{g_{\text{effective}} \cdot l} \]
where \( g_{\text{effective}} = g \sin \alpha \).

### Step 5: Substitute the Effective Gravitational Force
Substituting \( g_{\text{effective}} \):
\[ v = \sqrt{(g \sin \alpha) \cdot l} \]

### Step 6: Calculate the Required Velocity
To find the minimum velocity at the lowest point, we can express it as:
\[ v = \sqrt{g l \sin \alpha} \]

### Step 7: Consider the Condition for Reaching the Highest Point
For the particle to reach the highest point, its velocity at the lowest point must exceed a certain threshold. This threshold velocity can be derived from the energy conservation or dynamics of circular motion.

### Final Expression
The final expression for the minimum velocity at the lowest point for the particle to reach the highest point in its circular motion is:
\[ v \geq \sqrt{5 g l \sin \alpha} \]

### Conclusion
Thus, the velocity at the lowest point should exceed \( \sqrt{5 g l \sin \alpha} \) for the particle to successfully reach the highest point in its circular motion.

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