A stone of mass m sits at the bottom of a bucket. A tring is attached to the bucket and the bucket is made to move in circles. What is the minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket?

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To determine the minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket, we can analyze the forces acting on the stone at that point.

### Known Conditions:
1. Mass of the stone: $$ m $$
2. Gravitational acceleration: $$ g $$
3. Radius of the circular path (the height of the bucket): $$ r $$
4. Speed of the bucket at the highest point: $$ v $$

### Step-by-Step Solution:

1. **Forces Acting on the Stone:**
   At the highest point of the circular motion, the forces acting on the stone are:
   - The gravitational force acting downward: $$ F_g = mg $$
   - The normal force from the bucket acting downward: $$ F_n $$

The stone will remain in the bucket if the centripetal force required to keep it moving in a circle is provided by the gravitational force and the normal force.

2. **Centripetal Force Requirement:**
   The centripetal force required to keep the stone moving in a circle of radius $$ r $$ at speed $$ v $$ is given by:
   $$
   F_c = \frac{mv^2}{r}
   $$

3. **Force Balance at the Highest Point:**
   At the highest point, the centripetal force is provided by the sum of the gravitational force and the normal force:
   $$
   F_c = F_g + F_n
   $$
   Substituting the expressions for the forces, we have:
   $$
   \frac{mv^2}{r} = mg + F_n
   $$

4. **Minimum Condition:**
   The minimum speed occurs when the normal force $$ F_n $$ is zero (i.e., the stone is just about to lose contact with the bucket). Thus, we set $$ F_n = 0 $$:
   $$
   \frac{mv^2}{r} = mg
   $$

5. **Solving for Speed $$ v $$:**
   Rearranging the equation gives:
   $$
   v^2 = rg
   $$
   Taking the square root of both sides, we find:
   $$
   v = \sqrt{rg}
   $$

### Final Answer:
The minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket is:
$$
v = \sqrt{rg}
$$
The minimum speed required is $$ v = \sqrt{rg} $$.

A stone of mass m sits at the bottom of a bucket. A tring is attached to the bucket and the bucket is made to move in circles. What is the minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket?

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