A solid aluminum ingot weighs 89 N in air. (a) What is its volume? (See Table 13.1 . ) (b) The i

A solid aluminum ingot weighs 89 N in air. (a) What is its...

Key Concepts

Weight and Mass

Weight is the force due to gravity acting on an object’s mass and is calculated using the relationship F = mg, where F is the weight, m is the mass, and g is the gravitational acceleration. Understanding this concept is important when converting between weight and mass for further calculations, such as finding the volume of an object using its density.

Density

Density is defined as mass per unit volume and is a fundamental property of materials. It is used to relate the mass (or weight, when gravitational acceleration is known) of an object to its volume. Knowing an object's density allows one to compute its volume by rearranging the formula density = mass/volume.

Buoyant Force

Buoyant force is the upward force exerted by a fluid on a submerged object. According to Archimedes’ principle, this force is equal to the weight of the displaced fluid. It plays a key role in determining how much an object’s apparent weight is reduced when immersed in a fluid.

Apparent Weight

Apparent weight is the effective weight of an object when it is submerged in a fluid. It is calculated by subtracting the buoyant force from the object's actual weight. This concept is essential in understanding how fluids can reduce the net force (or tension) required to support an immersed object.

Transcript

00:01 This problem starts with an aluminum bar that has a weight of 89 newtons when in air.

00:06 We have two jobs.

00:07 First, we're going to find its volume, and then we're going to figure out what the tension in a rope would be if we held that aluminum bar by that rope so that it was suspended in water.

00:16 So first, let's figure out the volume.

00:18 That's pretty straightforward.

00:19 Now, we know that the density of aluminum is equal to 2 .7 times 10 to the third kilograms per cubic meter.

00:29 That's our density that we can look up.

00:32 And from this unit, we know that density is equal to mass over volume.

00:37 We also should already know that the weight of an object is equal to its mass times the acceleration due to gravity.

00:44 So what we can do with that information right away is we know that the weight of the bar is 89 newtons.

00:50 We need its mass if we're going to be able to do anything with that density, and this bar is on earth, where the acceleration due to gravity is 9 .8 meters per second squared.

01:00 Let's divide by 9 .8 on both sides.

01:04 And what we end up with, we know now that the mass of this aluminum bar is 9 .08163 kilograms is our mass.

01:16 Now that we know our mass, we can figure out the volume of this bar.

01:20 We can start with our density equation.

01:22 Density equals mass over volume.

01:24 What we can do is we can rearrange by multiplying by velocity on both sides and dividing by density.

01:30 To get, sorry, our volume on both sides, then dividing by density, to get that our volume is equal to mass over density.

01:39 We'll substitute in, that means that our volume is equal to, that 9 .08163 kilograms over our density to 0 .7 times 10 to the third, kilogram meters per meter cubed.

01:58 Now we can calculate that using our calculator, we end up with a volume of 0 .00364 meters cubed.

02:11 We can also say that we have 3 .364 liters, whether we want cubic meters or liters.

02:21 We can also both put these with two significant figures.

02:24 We were given a weight with two significant figures, so i can say that our volume is 0 .003.

02:33 3 4 cubic meters or i can say it's 3 .4 liters.

02:40 So we have found our volume.

02:42 That was our step one and our step two again is now we're going to find the tension in a rope if this bar was suspended in water.

02:50 So i think the easiest way to go about doing this is to draw a free body diagram with the forces on this bar.

02:57 So our box will represent our bar.

03:00 The primary force pulling down on it is going to be its weight and we luckily know that its weight is 89 newtons.

03:09 And then this bar is going to be at rest, suspended in water...

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