How to Calculate the Density of a Doughnut Submerged in Oil?

[MozAngeles]

Homework Statement

A popular brand of doughnut is made in an assembly-line fashion. First, the white, uncooked doughnut is dropped into hot vegetable oil ( ρ= 949 kg/m³) where it browns on one side as it cooks. It is then flipped over and browned on the other side. The final product still has a white stripe around the middle where the dough always stays out of the oil, as indicated in the figure .

Assuming the doughnut has a cylindrical shape of height H and diameter D, and that the height of the white stripe is 0.25H, find the density of the doughnut.

Homework Equations

F_buoyant=$$\rho$$Vg
F_buoyant=weight of doughnut
$$\rho$$=M/V
V_submerged=V_doughnut($$\rho$$_doughtnut/$$\rho$$_oil)

The Attempt at a Solution

I have not idea where to start.. the only conclusion I have come to is that the doughnut it floating so it's density has to be less than oil.
I know i must use most of these equations but I'm confused on which ones to use, considering i don't have numbers to plug in.
The whole white line in the center is throwing me off...

 

[anathema]

Thank you for your question. I can help you solve this problem by breaking it down step by step.

First, we need to understand what is happening in the assembly-line process. The doughnut is being dropped into hot vegetable oil, which has a density of 949 kg/m3. This means that the doughnut must have a density less than 949 kg/m3 in order to float in the oil.

Next, we need to consider the shape of the doughnut. It is a cylinder with a height of H and a diameter of D. This means that the volume of the doughnut can be calculated using the formula V = πr2h, where r is the radius (half the diameter) and h is the height.

Now, let's think about the white stripe in the middle of the doughnut. This is the part that does not touch the oil and therefore does not get cooked. The height of this white stripe is 0.25H, which means that the submerged part of the doughnut (the part in the oil) has a height of 0.75H.

Using the formula for density, ρ = m/V, we can rearrange it to solve for mass: m = ρV. This means that we need to find the volume of the doughnut in order to calculate its mass.

To find the volume of the doughnut, we need to subtract the volume of the white stripe from the total volume of the doughnut. The volume of the white stripe can be calculated using the formula for the volume of a cylinder (V = πr2h) with a height of 0.25H and a radius of D/2.

Now, we have the volume of the submerged part of the doughnut (Vdoughnut) and the volume of the white stripe (Vwhite stripe). We can use the formula Vsubmerged = Vdoughnut(ρdoughnut/ρoil) to find the density of the doughnut. We know that the density of the oil is 949 kg/m3, so we can plug that in and solve for ρdoughnut.

I hope this helps you solve the problem. Remember to always start with what you know and break the problem down into smaller steps. Good luck!