A cane filled with water is revolved in a vertical circle of radius 4 m and water just does not fall down. The time period of revolution will be –

Options

A. 4s
B. 2s
C. 1s
D. 6s

Explanation

To solve the problem of a cane filled with water being revolved in a vertical circle of radius 4 m, we need to determine the time period of revolution when the water just does not fall down. Here’s a step-by-step solution:

### Step 1: Understand the Forces at the Topmost Point
At the topmost point of the vertical circle, the forces acting on the water are:
- The gravitational force (weight) acting downwards, \( mg \)
- The centripetal force required to keep the water moving in a circle, which is provided by the gravitational force and the normal force from the cane.

For the water to just not fall, the normal force can be zero. Therefore, the centripetal force is entirely provided by the weight of the water.

### Step 2: Write the Equation for Centripetal Force
At the topmost point, the centripetal force is given by:
\[ \frac{mv^2}{r} = mg \]
Where:
- \( m \) is the mass of the water
- \( v \) is the velocity at the topmost point
- \( r \) is the radius of the circle (4 m)
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))

### Step 3: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{v^2}{r} = g \]
Rearranging gives us:
\[ v^2 = rg \]

### Step 4: Substitute the Values
Substituting \( r = 4 \, \text{m} \) and \( g = 9.8 \, \text{m/s}^2 \):
\[ v^2 = 4 \times 9.8 = 39.2 \]
Taking the square root to find \( v \):
\[ v = \sqrt{39.2} \approx 6.26 \, \text{m/s} \]

### Step 5: Calculate the Time Period
The time period \( T \) of one complete revolution is given by:
\[ T = \frac{2\pi r}{v} \]
Substituting \( r = 4 \, \text{m} \) and \( v \approx 6.26 \, \text{m/s} \):
\[ T = \frac{2\pi \times 4}{6.26} \]
Calculating this gives:
\[ T \approx \frac{25.13}{6.26} \approx 4 \, \text{seconds} \]

### Final Answer
The time period of revolution is approximately \( 4 \, \text{seconds} \).

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