Calculating Depth of Immersion of an Ice Cone in Water

[cooney88]

1. an ice cone 25mm high with a max radius of 15mm is floating apex downwards in a glass of water. if the ice cone has a mass of 5.3g to what depth will the cone be immersed? density of water is 1000kg/m^3



i know from the law of flotation that the weight of a floating body is equal to the weight of the fluid it displaces

so i should be using bouyancy = s1 x w / s

s1= density of object s = relative density of fluid and w is the wieght of the object but i can't seem to link this equation with anoter to find the depth.

maybe x/l x W?

The Attempt at a Solution

 

[LowlyPion]

You know the mass of the water that needs to be displaced.

Now the question arises what % of the volume of the cone will it immerse.

You might approach it as what is the size of a cone of displaced water that's equal to the mass of the original cone. Then that should tell you the depth of the ice cone that will be submerged shouldn't it? (You will need to keep the radius and the height in the same ratio.)

 

[nlightner]

:

Based on the given information, we can calculate the volume of the ice cone using the formula V = (1/3)πr^2h, where r = 15mm and h = 25mm. This gives us a volume of approximately 11.78 cm^3.

Next, we can use the given mass of 5.3g and the density of water (1000kg/m^3) to calculate the weight of the ice cone using the formula W = mg. This gives us a weight of 0.0053N.

Using the law of flotation, we know that the weight of the ice cone is equal to the weight of the water it displaces. We can calculate the weight of the displaced water using the density of water and the volume of the ice cone. This gives us a weight of 0.01178N.

Now, we can use the formula for buoyancy, B = ρVg, where ρ is the density of the fluid (in this case, water), V is the volume of the displaced fluid, and g is the acceleration due to gravity. Plugging in our values, we get B = (1000kg/m^3)(0.01178m^3)(9.8m/s^2) = 0.1156N.

Finally, we can use the formula for the depth of immersion, d = B/(ρwg), where ρ is the density of the object, w is the weight of the object, and g is the acceleration due to gravity. Plugging in our values, we get d = (1000kg/m^3)(0.0053N)/(1000kg/m^3)(9.8m/s^2) = 0.00054m or 0.54mm. This means the ice cone will be immersed to a depth of approximately 0.54mm in the water.