Lakes on Saturn's moon Titan are filled with liquid...

Lakes on Saturn's moon Titan are filled with liquid hydrocarbons-largely methane but also some ethane. Suppose scientists want to send a probe to explore Titan's largest lake (which is about the size of Earth's Lake Ontario). The probe is to be a cylinder $56.3 \mathrm{~cm}$ in diameter, with a mass of $135 \mathrm{~kg}$, and it's supposed to float with the long dimension vertical, half submerged and half above the lake surface. If the liquid in the lake has density $482 \mathrm{~kg} / \mathrm{m}^{3},$ what should be the probe's length?

Key Concepts

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. This force arises because the pressure exerted by a fluid increases with depth, resulting in a net upward push on the object. In floating problems, buoyancy is critical because an object floats when the buoyant force equals its weight, determining the volume of fluid displaced.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on a submerged object is equal to the weight of the fluid that the object displaces. This principle forms the basis for calculating the conditions under which an object will float, sink, or remain neutrally buoyant, by relating the object's weight to the density and volume of the displaced fluid.

Density

Density is a measure of mass per unit volume and is an essential property in determining how fluids interact with objects. In the context of buoyancy, both the density of the object and the density of the fluid are used to calculate the submerged volume required to generate enough buoyant force to support the object's weight.

Volume Displacement

Volume displacement refers to the amount of fluid that is pushed aside when an object is immersed in it. According to buoyancy principles, the weight of this displaced fluid must equal the weight of the object for it to float in equilibrium. This concept is used to determine how much of an object must be submerged to achieve a balance between gravitational and buoyant forces.

Transcript

00:05 Let's consider a scenario in which we have a cylinder, which serves as a probe on titan's moon, saturn, which is not entirely far -fetched as it goes.

00:17 I believe nasa is planning on sending a probe there in the year 2026 as part of its search for extraterrestrial life.

00:29 Now, let's think about what we're given, what we're looking for.

00:34 So we're given a variety of useful information, such as the diameter of the probe, about 56 .3 centimeters.

00:46 We're also given the mass of the probe, which is about 135 kilograms.

00:54 I'm also told that half is submerged, and we're told that the density of the liquid in which this probe exists is 482 kilograms being achieved.

01:16 Now what we're looking for is just the height of the probe for these conditions to be true.

01:34 Okay, so the key physical principle that you need to know here is archimedes principle, which is simply the idea that the mass of the displaced fluid should be equal to the mass of the object doing the displacing.

01:50 So the reason that makes sense is a force that moves subject downward.

01:56 Let's call it force of gravity, but all the forces here are gravitational forces.

02:01 That should cause the mass of our probe, the product of the mass for a probe, times gravity, or the gravitational force, to be equivalent to the mass of the displaced substance multiplied by gravity.

02:18 Now, you should find gravity cancels out here, so we just get massive debt, well, let's call it m -disp, because of us, is equal to the mass of the probe.

02:30 So given that information, we should be able, we should be well on our way to figuring how.

02:35 The height of the probe is.

02:36 So we're already given the mass of the probe.

02:39 But how will we find the mass in the displaced fluid? i think your way to it is probably easiest if you take the volume of the fluid being displaced, which we're told is half the volume of the probe.

02:52 And we multiply it by the density of the fluid.

02:57 So that should work out pretty easily.

02:59 We should find ourselves with units of meters here multiplied by kilograms of meters cubed, which would get us to a balance.

03:14 Now let's move on.

03:16 In order to solve this, we need to be able to solve for that value, which, well, it's just the equation for a cylinder.

03:27 So it's pi times r squared times height.

03:30 R here.

03:32 So i'll rather be looking r...