Water in a bucket is whirled in a vertical circle with a string attatched to it. The water does not fll down even when thebucket is inverted at the top of its path. We conclude that in this position

Options

A. `mg=(mv^2)/r`
B. mg is greater than `(mv^2)/r`
C. mg is not greater than `(mv^2)/r`
D. `mg is not less than (mv^2)/r

Answer

To solve the problem, we need to analyze the forces acting on the water in the bucket when it is at the top of its circular path. Here is a step-by-step solution:

### Step 1: Understand the Forces Acting on the Water
At the top of the circular path, two forces act on the water:
1. The gravitational force (weight) acting downward, which is \( mg \).
2. The centripetal force required to keep the water moving in a circle, which is provided by the tension in the string and the weight of the water.

### Step 2: Set Up the Equation for Forces
For the water to stay in the bucket without falling out, the net force acting on it must be directed towards the center of the circle. This means that the centripetal force must be equal to or greater than the gravitational force. Mathematically, we can express this as:

\[ F_{\text{centripetal}} \geq F_{\text{gravity}} \]

In terms of mass (\( m \)), velocity (\( v \)), and radius (\( r \)), the centripetal force is given by:

\[ \frac{mv^2}{r} \]

Thus, we can write the inequality:

\[ \frac{mv^2}{r} \geq mg \]

### Step 3: Simplify the Inequality
We can simplify the inequality by dividing both sides by \( m \) (assuming \( m \neq 0 \)):

\[ \frac{v^2}{r} \geq g \]

This tells us that for the water to not fall out of the bucket when it is inverted, the speed of the bucket must be such that the centripetal acceleration (\( \frac{v^2}{r} \)) is at least equal to the acceleration due to gravity (\( g \)).

### Step 4: Conclusion
From the analysis, we conclude that the water does not fall from the bucket when it is inverted at the top of its path because the centripetal force (due to the bucket's speed) is sufficient to counteract the gravitational force acting on the water.

### Final Answer
The conclusion we can draw is that the centripetal force must be greater than or equal to the gravitational force for the water to remain in the bucket when it is inverted.

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