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A bucket tied at the end of a 1.6m long ...

A bucket tied at the end of a `1.6m` long string is whirled in a verticle circle with constant speed. What should be the minimum speed so that the water from the bucket does not spill, when the bucket is at the highest position `(Takeg=10m//s^(2))`

A

`4m//sec`

B

`6.25m // sec`

C

`16m// sec`

D

None of the above

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of finding the minimum speed required for the water in a bucket to not spill when the bucket is at the highest position in a vertical circle, we can follow these steps: ### Step 1: Understand the Forces Acting on the Bucket At the highest point of the vertical circle, two forces act on the bucket: 1. The gravitational force (weight of the bucket and water), \( mg \), acting downwards. 2. The tension in the string (normal force), \( N \), also acting downwards. ### Step 2: Apply the Centripetal Force Requirement For the bucket to move in a circular path, the net force acting towards the center of the circle must provide the necessary centripetal force. At the highest point, the centripetal force is provided by the sum of the gravitational force and the tension in the string: \[ N + mg = \frac{mv^2}{r} \] Where: - \( m \) is the mass of the bucket and water, - \( v \) is the speed of the bucket, - \( r \) is the radius of the circle (length of the string). ### Step 3: Set the Condition for Minimum Speed To find the minimum speed at which the water does not spill, the tension \( N \) can be zero (the minimum condition). Thus, we can set \( N = 0 \): \[ mg = \frac{mv^2}{r} \] ### Step 4: Simplify the Equation We can cancel \( m \) from both sides (assuming \( m \neq 0 \)): \[ g = \frac{v^2}{r} \] ### Step 5: Solve for Speed \( v \) Rearranging the equation gives: \[ v^2 = g \cdot r \] Taking the square root: \[ v = \sqrt{g \cdot r} \] ### Step 6: Substitute the Given Values Given: - \( g = 10 \, \text{m/s}^2 \) - \( r = 1.6 \, \text{m} \) Substituting these values into the equation: \[ v = \sqrt{10 \cdot 1.6} = \sqrt{16} = 4 \, \text{m/s} \] ### Final Answer The minimum speed required so that the water does not spill when the bucket is at the highest position is: \[ \boxed{4 \, \text{m/s}} \]
To solve the problem of finding the minimum speed required for the water in a bucket to not spill when the bucket is at the highest position in a vertical circle, we can follow these steps: ### Step 1: Understand the Forces Acting on the Bucket At the highest point of the vertical circle, two forces act on the bucket: 1. The gravitational force (weight of the bucket and water), \( mg \), acting downwards. 2. The tension in the string (normal force), \( N \), also acting downwards. ### Step 2: Apply the Centripetal Force Requirement ...
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