The Michelson-Morley experiment (see Chapter 33 ) was a very...

The Michelson-Morley experiment (see Chapter 33 ) was a very precise optical experiment that helped pave the way for Einstein's theory of relativity. To isolate it from external vibrations, the entire experiment was mounted on a square slab of sandstone $0.30 \mathrm{~m}$ thick and $1.5 \mathrm{~m}$ on a side, with a mass of 1.7 tonnes $(1700 \mathrm{~kg})$. The slab, in turn, floated in a trough of liquid mercury (density 13.69 tonnes $/ \mathrm{m}^{3}$ ). What percentage of the slab's volume was below the surface of the mercury?

Key Concepts

Archimedes' Principle

This principle states that any object immersed in a fluid experiences an upward (buoyant) force equal to the weight of the fluid that the object displaces. It forms the basis for understanding why objects float or sink and is essential in calculating the submerged volume of an object in equilibrium with a fluid.

Density

Density is the measure of mass per unit volume of a substance. In buoyancy problems, comparing the density of the object to the density of the fluid determines the fraction of the object’s volume that will be submerged, as an object will displace a volume of fluid whose weight equals its own weight.

Equilibrium of Forces

In floating objects, equilibrium is achieved when the upward buoyant force exerted by the displaced fluid exactly balances the downward gravitational force on the object. This concept is central to determining how much of the object’s volume must be submerged to maintain balance, using the ratio of the object's density to the fluid's density.

Transcript

00:02 All right, so we're giving a scenario in which we have a slab immersed in mercury.

00:09 So there's a lot of interesting physics being discussed here.

00:12 We're just interested in this slab that is immersed in mercury.

00:20 That's it.

00:20 There's not other stuff we're really solving for.

00:24 We're just trying to figure out what fraction of the slab is submerged within the mercury.

00:36 So let's look at what we're given.

00:38 What kind of information we've given to solve this problem.

00:43 So we're all given the slab dimensions.

00:45 So it's a 1 .5 meter by 1 .5 meter square, which also has a third dimension of 0 .3 meters cubed.

01:05 Of just 0 .3 meters.

01:08 We'll solve for that in a second.

01:14 But of, i'll just label that volume, subscript slab.

01:20 What's the mass of this slab? it's about 1 .7 tons.

01:28 Metric tons.

01:31 One ton is equal to 1 ,000 kilograms.

01:39 So let's go on my limb here and say, this is 1 ,700 kilograms.

01:46 That's fine.

01:47 We also are given the row of the mercury.

01:51 Just to know row m.

01:53 That equals 13 .69 tons per meters cubed.

02:01 Should equal 13690 tons i'm going to have space here doesn't seem like i'll have much more 1 ,3690 tons what am i talking about? it's a meter skewed.

02:27 My bad.

02:30 So it's 13 ,690 kilograms.

02:40 What are we trying to find? we're just trying to find the map fraction of the slab that is immersed.

02:53 For the percentage, we can express our answer in percentage too.

03:15 All right.

03:20 So we'll call the displaced mercury mass.

03:30 We're going to say that's equal to the mass of the slab.

03:33 Why can we say that? this is archimedes principle.

03:36 Archimedes principle indicates that the mass of the displaced liquid will be equal to the mass of the object that is displacing.

03:49 The reason we know this is because the gravitational force downwards of the sloth.

03:54 But there's also a force generated upwards as the molecules try to find their way around this and accommodate this object that is being submerged.

04:06 Let's go ahead and clarify this just a little bit.

04:13 Let's say this is the mass of the slab.

04:17 Let's just call this mdisp.

04:20 I don't think i'm going to use this notation later, maybe about that would also be the product of gravity...

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