Finding Volume: Stone Submerged In Water - Physics

Hey guys! Let's dive into a cool physics problem involving buoyancy, density, and a dynamometer. We're going to figure out the volume of a stone submerged in water. It's a classic physics scenario that combines a few important concepts, so stick with me, and we'll break it down step by step!

Setting Up the Problem

So, here’s the deal: we've got a stone with a mass of 0.2 kg. This stone is chilling underwater, totally submerged. Now, it's not just sinking to the bottom because it's being held up a bit by a dynamometer. Think of a dynamometer like a fancy scale that measures force. In this case, it reads 1.5 N (Newtons). We also know the density of the water is 1000 kg/m³, and we're using a gravitational acceleration of 10 m/s². Our mission? To find the volume of this sneaky stone.

Understanding the Forces at Play

Before we start crunching numbers, let's get a handle on the forces acting on the stone. There are three main players here:

1. Weight (W): This is the force pulling the stone down due to gravity. We calculate it as W = mg, where m is the mass and g is the acceleration due to gravity.
2. Buoyant Force (B): This is the upward force exerted by the water on the stone. It's what makes things feel lighter underwater. The buoyant force is equal to the weight of the water displaced by the stone. We calculate it as B = ρVg, where ρ is the density of the water, V is the volume of the stone (which is what we want to find), and g is the acceleration due to gravity.
3. Tension (T): This is the force exerted by the dynamometer pulling upwards. It's the reading we get from the dynamometer, which is 1.5 N.

The Equilibrium Condition

Since the stone is in equilibrium (meaning it's not moving up or down), all the forces acting on it must balance out. This means the sum of the upward forces equals the sum of the downward forces. In our case, the upward forces are the buoyant force (B) and the tension (T), and the downward force is the weight (W). So, we can write the equation:

B + T = W

Now, let's express each of these forces in terms of the quantities we know.

Calculating the Forces

Alright, let's calculate each force individually.

1. Weight (W)

The weight of the stone is pretty straightforward:

W = mg = 0.2 kg * 10 m/s² = 2 N

So, the stone weighs 2 Newtons.

2. Buoyant Force (B)

We don't know the buoyant force directly yet, but we know it's related to the volume of the stone, which is what we're trying to find. We'll leave it as B for now and solve for it later using our equilibrium equation.

3. Tension (T)

The tension is given directly by the dynamometer reading:

T = 1.5 N

Solving for the Buoyant Force

Now we can plug the values we found for W and T into our equilibrium equation:

B + T = W B + 1.5 N = 2 N

Solving for B, we get:

B = 2 N - 1.5 N = 0.5 N

So, the buoyant force acting on the stone is 0.5 Newtons. This means the weight of the water displaced by the stone is 0.5 N.

Finding the Volume of the Stone

Now comes the fun part – using the buoyant force to find the volume of the stone! We know that the buoyant force is given by:

B = ρVg

Where:

• B = Buoyant force (0.5 N)
• ρ = Density of water (1000 kg/m³)
• V = Volume of the stone (what we want to find)
• g = Acceleration due to gravity (10 m/s²)

Let's rearrange the equation to solve for V:

V = B / (ρg)

Now, plug in the values:

V = 0.5 N / (1000 kg/m³ * 10 m/s²) V = 0.5 N / 10000 kg/(m²s²) V = 0.00005 m³

To make this a bit easier to grasp, let's convert it to cubic centimeters (cm³). Remember that 1 m³ = 1,000,000 cm³:

V = 0.00005 m³ * 1,000,000 cm³/m³ = 50 cm³

So, the volume of the stone is 50 cubic centimeters.

Wrapping Up

There you have it! By understanding the forces acting on the stone and using the principle of equilibrium, we were able to determine its volume. Remember, the key to these problems is breaking them down into smaller, manageable steps and understanding the underlying physics principles. This problem beautifully illustrates how buoyancy, weight, and tension interact in a submerged object.

Keep practicing, and you'll become a physics whiz in no time! Understanding the buoyant force is crucial; it's the force that opposes the weight of an object immersed in a fluid. Calculating the weight accurately is also vital, as it's the force pulling the object downwards. By equating these forces in equilibrium, we can solve for unknowns like the volume of the stone. Remember that the dynamometer reading gives us the tension, which is another piece of the puzzle.

Understanding these concepts is beneficial not just for exams but also for real-world applications. Engineers use these principles when designing ships, submarines, and other underwater structures. So, keep exploring and keep learning, guys!

Real-World Application

Understanding the principles behind this problem helps us grasp real-world phenomena. For instance, when designing a boat, engineers need to consider the buoyant force to ensure the boat floats. Similarly, submarines use ballast tanks to control their buoyancy, allowing them to submerge and resurface. The concept of equilibrium is fundamental in architecture, ensuring buildings remain stable under various loads.

In the medical field, understanding fluid dynamics is crucial for designing artificial organs and blood pumps. In environmental science, these principles help predict how pollutants disperse in water bodies. So, the knowledge gained from solving problems like this is transferable and invaluable across various disciplines.

Advanced Concepts

To deepen your understanding, let’s touch upon some advanced concepts related to this problem. The Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid it displaces. This principle is the cornerstone of buoyancy calculations. The density of the fluid and the volume of the submerged object are key factors in determining the buoyant force. Additionally, the shape of the object can affect the fluid dynamics around it, influencing the buoyant force.

Moreover, the concept of center of buoyancy is essential for understanding the stability of floating objects. The center of buoyancy is the centroid of the displaced fluid. If the center of gravity of the object is below the center of buoyancy, the object is stable. If it’s above, the object is unstable and may topple over. Understanding these advanced concepts enhances your ability to solve more complex problems involving fluid dynamics and buoyancy.

I hope this helps clarify how to approach problems involving buoyancy, equilibrium, and fluid dynamics. Keep up the great work, and always stay curious! You've got this!