Density Calculation: Silver Ornament In Water (SI Units)

Hey guys! Today, we're diving into a fun physics problem: figuring out the density of a silver ornament. This isn't just some abstract calculation; it's a practical application of understanding density and displacement, something you might encounter in real-world situations. We'll break it down step-by-step, so you can see exactly how it's done. So, let's get started and understand how to calculate the density of a silver ornament using the principles of buoyancy and displacement.

Understanding the Problem

Before we jump into the calculations, let's make sure we understand the problem. We've got a silver ornament that weighs 0.105 kg in air. When we dunk it in water, it appears to weigh only 95 g (which is 0.095 kg). Why the difference? Buoyancy! The water is pushing up on the ornament, making it seem lighter. This buoyant force is equal to the weight of the water displaced by the ornament. Our mission, should we choose to accept it, is to find the density of this silver ornament in kg/m³, the standard SI unit for density.

Density, at its core, is a measure of how much stuff (mass) is packed into a certain amount of space (volume). Think of it this way: a kilogram of feathers takes up way more space than a kilogram of lead. Lead is denser because it has more mass crammed into the same volume. The formula we'll be using is pretty straightforward:

Density = Mass / Volume

So, to find the density of our silver ornament, we need to figure out its mass (which we already have) and its volume. The trick here is that we don't have the volume directly, but we can figure it out using the information about how the ornament behaves in water. This involves the principle of Archimedes, which is a fundamental concept in fluid mechanics. By understanding this principle, we can effectively solve the problem and determine the density of the silver ornament.

Step 1: Finding the Buoyant Force

Okay, first things first: let's calculate the buoyant force. Remember, the buoyant force is the difference between the weight of the ornament in air and its apparent weight in water. This is a crucial step in determining the density, as it allows us to link the weight difference to the volume of water displaced, which in turn gives us the volume of the ornament.

Buoyant Force = Weight in Air - Weight in Water

We know the weight in air is 0.105 kg. To get the weight in Newtons (the standard unit of force), we multiply by the acceleration due to gravity, which is approximately 9.8 m/s².

Weight in Air = 0.105 kg * 9.8 m/s² = 1.029 N

The apparent weight in water is 95 g, which is 0.095 kg. Let's convert that to Newtons too:

Weight in Water = 0.095 kg * 9.8 m/s² = 0.931 N

Now we can find the buoyant force:

Buoyant Force = 1.029 N - 0.931 N = 0.098 N

So, the buoyant force acting on the silver ornament is 0.098 Newtons. This value represents the upward force exerted by the water on the ornament, and it's directly related to the amount of water displaced. Understanding this buoyant force is key to unlocking the ornament's volume, which is the next piece of the puzzle in calculating its density. Let's move on to the next step!

Step 2: Calculating the Volume of Displaced Water

Here's where things get a little clever. Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the object. In our case, the fluid is water. We know the buoyant force (0.098 N), so we can work backward to find the volume of water displaced. This is a crucial step because the volume of the displaced water is equal to the volume of the silver ornament itself. This connection allows us to indirectly measure the ornament's volume, which is essential for calculating its density.

First, we need to remember the relationship between weight, mass, and gravity:

Weight = Mass * Gravity

We know the buoyant force (which is the weight of the displaced water), and we know gravity (9.8 m/s²). So, we can find the mass of the displaced water:

Mass of Displaced Water = Buoyant Force / Gravity Mass of Displaced Water = 0.098 N / 9.8 m/s² = 0.01 kg

Now, we know the mass of the displaced water is 0.01 kg. To find the volume, we need to use the density of water. The density of water is approximately 1000 kg/m³.

Density = Mass / Volume

Rearranging to solve for Volume:

Volume = Mass / Density Volume = 0.01 kg / 1000 kg/m³ = 0.00001 m³

So, the volume of water displaced (and therefore the volume of the silver ornament) is 0.00001 m³, which is also equal to 1 x 10^-5 m³. We've now successfully determined the volume of the silver ornament by leveraging the principle of buoyancy and the properties of water. This was a critical step, as we now have all the necessary information to calculate the ornament's density.

Step 3: Determining the Density of the Silver Ornament

Alright, we're in the home stretch! We've got the mass of the silver ornament (0.105 kg) and we've figured out its volume (0.00001 m³). Now, we can finally calculate the density using our trusty formula:

Density = Mass / Volume

Plugging in the values:

Density = 0.105 kg / 0.00001 m³ = 10500 kg/m³

So, the density of the silver ornament is 10500 kg/m³. That's a pretty dense piece of silver! This result gives us a quantitative understanding of how much mass is packed into each unit of volume within the ornament. It's a characteristic property of the material itself, and knowing the density can help us identify the substance and its purity.

Conclusion: Density Solved!

And there you have it! We successfully calculated the density of the silver ornament. We took a seemingly tricky problem and broke it down into manageable steps. We used the principles of buoyancy, Archimedes' principle, and the density of water to find our answer. It's a great example of how physics concepts can be applied to solve real-world problems. This journey from understanding the initial conditions to arriving at the final density value showcases the power of problem-solving in physics.

Remember, the key takeaways here are:

• Buoyant force is the difference between the weight in air and the weight in a fluid.
• Archimedes' principle tells us that the buoyant force equals the weight of the displaced fluid.
• Density is mass divided by volume.

Hopefully, this step-by-step explanation has made the process clear and easy to follow. If you encounter similar problems in the future, remember to break them down into smaller steps and utilize the fundamental principles of physics. Keep practicing, and you'll become a density-calculating pro in no time! Now, go forth and conquer more physics challenges!