A cylindrical container is 30cm in diameter. If is filled at rest by 20 cm of fresh water, and then spun at 30 rpm. after state of solid body rotation is achieved, what is the difference in level between the rim and the center
To find the difference in level between the rim and the center of the cylindrical container after achieving solid body rotation, we can consider the concept of the centrifugal force acting on the water.
First, let's calculate the height of the water at the rim of the container. The diameter of the container is 30 cm, so the radius (r) is half of that, which is 15 cm or 0.15 meters.
Next, we need to find the centrifugal force experienced by the water at the rim due to the spinning motion. The centrifugal force can be calculated using the formula:
Centrifugal force = mass × radius × angular velocity squared
In this case, the mass of the water is given by its density (which is the same as fresh water, around 1000 kg/m³) multiplied by its volume. The volume can be calculated by multiplying the cross-sectional area of the container (πr²) by its height. Since the height of water at the rim is the same as the height of the container (20 cm or 0.2 meters), the volume of water is (π × 0.15² × 0.2) cubic meters.
Now, let's plug in the values into the formula to calculate the centrifugal force at the rim:
Centrifugal force at the rim = (density × volume) × radius × angular velocity squared
Next, we need to find the acceleration due to the centrifugal force at the rim, which can be calculated by dividing the centrifugal force by the mass of the water. The mass can be calculated by multiplying the density by the volume of water.
Finally, we can calculate the difference in level between the rim and the center by using the equation:
Difference in level = (acceleration due to centrifugal force) / (acceleration due to gravity)
Now, plug in the values and perform the calculations to find the difference in level between the rim and the center of the cylindrical container after achieving solid body rotation.