Final answer:
To find the minimum speed of the pail at the top of the vertical circle, we can set the centripetal force equal to the weight of the water in the pail. Solving for the angular velocity gives us the minimum speed of the pail at the top of the circle.
Step-by-step explanation:
To find the minimum speed of the pail at the top of the vertical circle, we need to consider the forces at play. At the top of the circle, the forces acting on the pail are the weight of the water and the centripetal force.
The centripetal force required to keep the water in the pail at the top of the circle without spilling is equal to the weight of the water. We can calculate the centripetal force using the formula:
Centripetal force = mass x radius x angular velocity^2
Since the water in the pail does not spill, the centripetal force is equal to the weight of the water, which is equal to mass x acceleration due to gravity. We can set these two equations equal to each other and solve for the minimum angular velocity required:
mass x radius x angular velocity^2 = mass x acceleration due to gravity
Dividing both sides of the equation by the mass, the mass cancels out:
radius x angular velocity^2 = acceleration due to gravity
Substituting the known values (radius = 0.80 m and acceleration due to gravity = 9.8 m/s^2), we can solve for the angular velocity:
0.80 m x angular velocity^2 = 9.8 m/s^2
angular velocity^2 = 9.8 m/s^2 / 0.80 m
angular velocity^2 = 12.25 (m/s^2)^2
Taking the square root of both sides to find the angular velocity:
angular velocity = sqrt(12.25) m/s
angular velocity ≈ 3.50 m/s
Therefore, the minimum speed of the pail at the top of the circle is approximately 3.50 m/s.
Forces at play:
- Weight of the water
- Centripetal force