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, we need to consider the effect of centrifugal force caused by the rotation.
Step 1: Calculate the radius of the cylindrical container:
The diameter of the container is 30 cm, so the radius (r) is half of that, which is 15 cm.
Step 2: Calculate the angular velocity (ω):
The given information states that the container is spun at 30 rpm (revolutions per minute). To convert this to radians per second (which is needed for calculations), we need to multiply it by 2π/60.
ω = (30 rpm) x (2π/60) = π rad/s.
Step 3: Calculate the difference in water level:
The difference in water level is equal to the height gained due to the centrifugal force acting on the water.
Centrifugal force (F) acting on an object at a distance (r) from the center of rotation is given by:
F = m * r * ω^2,
where m is the mass of the object, r is the distance, and ω is the angular velocity.
In this case, the object is a column of water.
The mass of the water can be calculated by multiplying its volume by its density. The volume of the water can be calculated by multiplying the cross-sectional area of the container by its height.
The cross-sectional area (A) of the container is given by:
A = π * r^2,
where r is the radius of the container.
The height (h) of the water is given as 20 cm.
Thus, the mass of the water (m) is given by:
m = A * h * density,
where density is the density of water (approximately 1000 kg/m^3).
To convert the mass to kilograms, we divide by 1000.
Once we have the mass of the water (m), we can calculate the centrifugal force (F) acting on it using the formula above.
The difference in water level is then given by:
difference in level = F / (m * g),
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
By following these steps and performing the calculations, you can find the difference in level between the rim and the center of the cylindrical container.