Calculating Canoe Mass: A Physics Problem

Hey everyone! Today, we're diving into a fun physics problem that involves gravity, canoes, and a little bit of math. The core of this problem revolves around understanding how the gravitational force works between two objects, in this case, a couple of canoes floating in the water. We'll be using the fundamental principles of Newton's Law of Universal Gravitation to figure out the mass of the lighter canoe. So, grab your calculators, and let's get started!

Understanding the Problem: Gravity's Canoe Edition

Okay, so the setup is pretty straightforward. We have two canoes chilling in the water, and they're 1,500 meters apart. We know that gravity is at play here, and there's a tiny gravitational force pulling these canoes towards each other – it's tiny because, well, canoes aren't exactly massive objects. The problem tells us that this gravitational force is 2.378 × 10⁻¹³ N (Newtons).

Now, here's the twist: one canoe is twice as massive as the other. This is a crucial piece of information that we'll use to solve the problem. What we need to find is the mass of the lighter canoe. This means we'll need to employ some algebra to unravel this mystery.

To really grasp what we're doing, let's zoom in on Newton's Law of Universal Gravitation. This law states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula looks like this: F = G * (m1 * m2) / r². Where:

• F is the gravitational force.
• G is the gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²).
• m1 and m2 are the masses of the two objects.
• r is the distance between the centers of the two objects.

We know F, G, and r. The trick is figuring out m1 and m2 given the relationship that one canoe's mass is twice the other. We'll use this info to create an equation that we can solve. Keep in mind that solving these problems involves careful application of the formula and understanding of the relationship between variables. You've got this!

Setting Up the Equation: Putting the Pieces Together

Alright, let's break down how to set up the equation to solve this canoe mass conundrum. The key is to define our variables clearly and use the information we have to create a solvable equation.

First, let's denote the mass of the lighter canoe as m. Since the other canoe is twice as massive, its mass will be 2m. The distance between the canoes (r) is 1,500 meters, and the gravitational force (F) is 2.378 × 10⁻¹³ N. The gravitational constant (G) is a standard value: 6.674 × 10⁻¹¹ N⋅m²/kg².

Now, let's plug these values into Newton's Law of Universal Gravitation formula: F = G * (m1 * m2) / r². Substituting our variables:

• 2.378 × 10⁻¹³ = 6.674 × 10⁻¹¹ * (m * 2m) / (1500)²

Notice that we've replaced m1 and m2 with m and 2m, respectively. This is a very critical step. This gives us an equation with only one unknown variable, m, which represents the mass of the lighter canoe.

Our equation, at this point, captures the entire essence of the problem. It is the bridge between the given data and the answer we are trying to find. Solving this equation is where the magic happens; we'll isolate m to find its value. Remember to be meticulous with your calculations, and don’t forget to keep track of the units, which can help ensure you're on the right track! The setup is most of the battle; the rest is just arithmetic. Let's solve it!

Solving for the Mass: Unveiling the Answer

Okay, time to crunch some numbers and solve for m, which is the mass of the lighter canoe. Let's take the equation we set up and simplify it step-by-step to isolate m. Our equation is:

• 2. 378 × 10⁻¹³ = 6.674 × 10⁻¹¹ * (m * 2m) / (1500)²

First, simplify the right side of the equation. Combine the masses and calculate the square of the distance:

• (1500)² = 2,250,000
• m * 2m = 2m²

So, the equation becomes:

• 2.378 × 10⁻¹³ = 6.674 × 10⁻¹¹ * (2m²) / 2,250,000

Next, multiply 6.674 × 10⁻¹¹ by 2:

• 6.674 × 10⁻¹¹ * 2 = 1.3348 × 10⁻¹⁰

Now our equation looks like this:

• 2. 378 × 10⁻¹³ = 1.3348 × 10⁻¹⁰ * m² / 2,250,000

To isolate m², multiply both sides by 2,250,000:

• 2.378 × 10⁻¹³ * 2,250,000 = 0.000000053505

• 1. 3348 × 10⁻¹⁰ * m²

So the equation transforms to:

• 0.000000053505 = 1.3348 × 10⁻¹⁰ * m²

Divide both sides by 1.3348 × 10⁻¹⁰:

• 0. 000000053505 / 1.3348 × 10⁻¹⁰ = 0.401

• m² = 0.401

To find m, take the square root of both sides:

• m = √0.401

• m ≈ 0.63 kg

Therefore, the mass of the lighter canoe is approximately 0.63 kg. Ta-da! See, physics can be fun, right? Remember, we started with a problem, applied a law, and now we have an answer. This systematic approach is fundamental in science.

Final Thoughts and Key Takeaways

Alright, folks, we've successfully navigated the waters of this physics problem! We calculated the mass of the lighter canoe using Newton's Law of Universal Gravitation. This problem shows how we can apply physics to everyday scenarios and use mathematical tools to solve real-world questions.

Here are the key takeaways:

• Understanding the Formula: A firm grasp of the formula and the variables involved is critical.
• Setting up the Equation: The way you structure the equation determines your success in solving the problem.
• Careful Calculations: Accuracy in calculations is key to reaching the correct answer.

By following these steps and understanding the principles of physics, you too can solve problems like this with ease. Physics might seem daunting, but it's really just a way of understanding how the world works, and hopefully, you guys had fun with this one. Keep practicing, keep exploring, and who knows, maybe you'll be calculating gravitational forces between canoes in your spare time! Until next time, keep those scientific minds engaged. Thanks for hanging out, and don't be afraid to try some more practice problems. It's the best way to get better. Cheers!