A hunk of aluminum is completely covered with a gold shell to form an ingot of weight 45.0 N . W

step 1 Okay, so in this problem we have a hunk of aluminium. So that's our aluminum that is completely

A hunk of aluminum is completely covered with a gold shell...

Key Concepts

Density and Mass?Volume Relationship

Density is defined as mass per unit volume and is a key property in determining how much volume a given mass occupies. In problems involving composite materials or multi?material objects, understanding each component’s density allows one to relate the mass and the volume, which is crucial when combined with buoyancy considerations.

Composite Object Analysis

When dealing with objects made of more than one material, the overall properties (such as mass, volume, and weight) are determined by summing the contributions from each component. By isolating the contributions according to density, one can analyze how each material affects factors like displaced volume under submersion, leading to a solution for the individual weights or masses of the components.

Archimedes' Principle

This principle states that any object, wholly or partially immersed in a fluid, experiences an upward force equal to the weight of the fluid displaced by it. It provides the foundation for analyzing buoyant forces and is essential for understanding changes in apparent weight when objects are submerged in fluids.

Buoyant Force

Related to Archimedes' Principle, buoyant force is the upward force exerted by a fluid on a submerged object. It is calculated as the product of the displaced fluid’s volume, the fluid density, and gravitational acceleration. This concept explains why submerged objects appear lighter and how differences in fluid displacement can be used to deduce properties of an object.

Transcript

00:01 Okay, so in this problem we have a hunk of aluminium.

00:06 So that's our aluminum that is completely covered by a gold shell.

00:12 So that is the gold shell.

00:15 We know the weight of this ingot.

00:21 We have a string in here that is attached to a balance.

00:27 And we know that this ingot is submerged into water.

00:31 So we have here a recipient with water.

00:39 Okay? so first of all, what are the forces involving this problem? well, let's see.

00:48 We know that we have the weight of this ingot pointing downwards.

00:56 We have a buoyant force and we have a tension.

00:59 So we have a tension in here.

01:01 Let's call this t.

01:03 And we have a buoyant force.

01:05 So let's call fb nice okay so first of all what is the definition of the buoyant force so let's remember this the buoyant force fb is just the density of the liquid which in this case is water the volume submerged times the gravitational acceleration okay so just to remember that is the buoyant force now let's solve the problem what we must do to solve this problem? first of all, let's write down the newton second law for this system.

01:50 So we have that buoyant force plus the tension on the string minus the weight of the ingot is going to be equal zero.

02:06 Therefore, we can say that using the buoyant force that we have in here, we can say that of the water, they multiplies the volume, g is going to be equal the weight.

02:26 Let's put on different w here.

02:30 So that is the weight minus the tension.

02:35 Okay? so we know that the volume that is submerged, let's write this here.

02:49 The volume that is submerged is just the weight of the ingot minus the tension divided by, let's see, the density of the water times g.

03:07 Therefore we can calculate this volume.

03:09 This volume is going to be 45 minus the tension, which is 39, divided by the density of the water, 1 ,000, that multiplies the gravity acceleration of 9 .8.

03:27 So calculating all this, we have first of all a volume submerge of 6 .12 times 10 to the minus 4 middles cubics, but why we are calculating the volume if we want to calculate what is the mass of the gold shell.

03:56 Actually not the mass, the weight of the gold shell.

04:01 We are calculating the volume because let's continue here the equation.

04:09 Because we know that the total mass of the ingot, let's put here mt, the total mass of the ingot.

04:17 Is going to be equal the mass of the gold plus the mass of the aluminum.

04:27 And we can use the density relation here to describe this total mass as the ho of the gold times the volume of the gold plus the ho of aluminum times the volume of aluminum...

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