Density Calculation: Parallelepiped Dimensions & Force

Hey guys! Today, we're diving into a classic physics problem: figuring out the density of an object. We've got a parallelepiped (fancy word for a box!) with specific dimensions, and we know the force it exerts when hanging from a dynamometer. Sounds like fun, right? Let's break it down step by step.

Understanding the Problem

Okay, so the core of our mission is to determine the density of the material that makes up this parallelepiped. We know its dimensions are 8cm x 4cm x 2cm, and when it's suspended from a dynamometer, it shows a force of 5N. This force, guys, is actually the weight of the parallelepiped, which is super important for our calculations. Density, remember, is a measure of how much "stuff" is packed into a certain space, and it’s defined as mass per unit volume. So, we need to find both the mass and the volume of the parallelepiped to calculate its density.

To really nail this, we need to remember a few key physics concepts. First off, we'll use the dimensions to figure out the volume of the parallelepiped. Since it’s a regular shape, this will be straightforward. Next, we'll use the force measured by the dynamometer (the weight) to find the mass. Here, we’ll need to recall the relationship between weight, mass, and gravitational acceleration. Finally, with both mass and volume in hand, we can confidently calculate the density. The formula we'll be leaning on is Density = Mass / Volume (ρ = m/V). Let’s get started and crunch some numbers, making sure we keep our units consistent throughout the process. This problem isn't just about plugging numbers into a formula; it’s about understanding the relationships between physical quantities and applying the right concepts to solve a real-world (or at least, a textbook-world!) problem.

Step 1: Calculate the Volume

Alright, let's kick things off by calculating the volume of our parallelepiped. This is a crucial first step because, as we know, density is mass divided by volume. For a parallelepiped, the volume calculation is super straightforward – we just multiply its length, width, and height together. In our case, the dimensions are given as 8cm x 4cm x 2cm. So, let’s plug those numbers in: Volume = 8cm * 4cm * 2cm = 64 cubic centimeters (cm³). But hold on a second! To keep things consistent with standard units in physics, we need to convert this volume from cubic centimeters to cubic meters (m³). Remember, guys, the standard unit for volume in the International System of Units (SI) is the cubic meter. There are 100 centimeters in a meter, so there are (100cm)³ = 1,000,000 cm³ in 1 m³. To convert 64 cm³ to m³, we divide by 1,000,000: 64 cm³ = 64 / 1,000,000 m³ = 0.000064 m³. So, the volume of our parallelepiped is 0.000064 m³. This conversion is essential because we'll be using the standard unit of force (Newtons) and gravitational acceleration (m/s²) in the next steps, and we want to make sure all our units play nicely together. Getting the volume right, in the correct units, sets us up for a smooth calculation of the density later on. Trust me, paying attention to units is a lifesaver in physics problems! It prevents a lot of common mistakes and makes sure our final answer makes sense.

Step 2: Determine the Mass

Now that we've got the volume sorted out, it's time to figure out the mass of the parallelepiped. This step is where the force reading from the dynamometer comes into play. Remember, the dynamometer indicates a force of 5N, which is actually the weight of the parallelepiped. Weight, in physics terms, is the force exerted on an object due to gravity. The relationship between weight (W), mass (m), and gravitational acceleration (g) is given by the formula: W = m * g. We know the weight (W = 5N), and we also know the standard value for gravitational acceleration on Earth, which is approximately g = 9.8 m/s². What we're after is the mass (m), so we need to rearrange the formula to solve for m: m = W / g. Let's plug in the values: m = 5N / 9.8 m/s² ≈ 0.51 kilograms (kg). So, the mass of our parallelepiped is approximately 0.51 kg. It’s really important to use the standard units here (Newtons for force, meters per second squared for gravitational acceleration) to get the mass in kilograms, which is the standard unit for mass in the SI system. This step links the force measurement directly to the mass, which is a key piece of information for calculating density. By understanding the relationship between weight and mass, we can accurately determine the mass of the parallelepiped from the dynamometer reading. This careful attention to the underlying physics principles is what makes problem-solving in physics so rewarding, guys!

Step 3: Calculate the Density

Alright, we're in the home stretch now! We've successfully calculated both the volume and the mass of the parallelepiped, which means we have all the pieces we need to determine its density. As we discussed earlier, density (ρ) is defined as mass (m) divided by volume (V): ρ = m / V. We found the mass to be approximately 0.51 kg and the volume to be 0.000064 m³. So, let's plug these values into our formula: ρ = 0.51 kg / 0.000064 m³ ≈ 7968.75 kg/m³. Therefore, the density of the material from which the parallelepiped is made is approximately 7968.75 kilograms per cubic meter. Guys, that's a pretty dense material! To give you a sense of scale, the density of water is 1000 kg/m³, so our parallelepiped is significantly denser than water. This high density suggests that the material could be something like iron or steel. Calculating the density is the final step in solving our problem, and it showcases how different physical quantities are interconnected. By carefully working through each step – calculating the volume, determining the mass, and finally applying the density formula – we've been able to successfully characterize a property of the material. This process highlights the power of physics in understanding the world around us, making it super interesting, right?

Conclusion

So, there you have it! We've successfully determined the density of the parallelepiped. We started with the dimensions of the object and the force it exerted, and through careful calculations and a good understanding of physics principles, we arrived at the answer. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. First, we found the volume using the given dimensions. Then, we used the force measurement and the concept of weight to calculate the mass. Finally, we combined these two results using the density formula to get our final answer. The density of the material is approximately 7968.75 kg/m³, indicating it's a pretty dense substance. This exercise, guys, is a great example of how physics can be used to analyze and understand the properties of objects around us. Density is a fundamental property of matter, and knowing how to calculate it from basic measurements is a valuable skill. Keep practicing these problem-solving techniques, and you'll become a physics whiz in no time!