Horizontal Acceleration Calculation: Physics Explained

Hey guys! Let's dive into a classic physics problem: "A force of 100 newtons is applied to a box at an angle of 36° with the horizontal. If the mass of the box is 25 kilograms, what is the horizontal acceleration of the box?" This is a great example of how forces, angles, and mass interact to cause motion. We're going to break down the problem step-by-step so you can totally nail it. We will be using the key concepts of Newton's Second Law and trigonometry to find the horizontal acceleration. Don't worry if these terms sound a bit intimidating; we'll explain everything in a way that's easy to understand. So, grab your calculators and let's get started on figuring out the horizontal acceleration, shall we?

Understanding the Problem: Forces and Motion

Alright, before we jump into the calculations, let's make sure we've got a solid grasp of what's going on. In this scenario, we have a box sitting on a surface, and someone is pushing it with a force of 100 Newtons. Now, the force isn't just a straight push; it's at an angle of 36 degrees relative to the ground. This angle is super important because it means only part of the applied force is actually moving the box horizontally. The other part of the force is pushing down on the box, but we're only interested in the horizontal movement for now. The mass of the box is 25 kilograms, which tells us how much the box resists changes in its motion (inertia). To solve the problem, we need to find the horizontal component of the applied force, which is the part that's directly causing the box to accelerate horizontally. This horizontal component is what we will use to calculate the acceleration. The key to solving this type of problem is to break down the force into its horizontal and vertical components. This will allow us to use Newton's Second Law to solve for the acceleration. So, we'll need to use some basic trigonometry, specifically the cosine function, to find this component. This step is critical because it connects the applied force with the actual force responsible for the box's horizontal movement. Then, we can use the horizontal force and the mass of the box to find the acceleration using Newton's Second Law of Motion. Remember that the horizontal acceleration is what we are looking for.

The Role of Newton's Second Law

Newton's Second Law of Motion is the superstar of this problem. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). Where F represents force, m represents mass, and a represents acceleration. In our case, the horizontal component of the applied force is the 'F', the mass of the box is 'm', and the horizontal acceleration we want to find is 'a'. We rearrange the formula to solve for acceleration, the formula then becomes a = F/m. This law helps us understand the relationship between force, mass, and acceleration, and it's essential for solving problems like this. Without Newton's Second Law, we wouldn't be able to relate the force acting on the box to its movement. So, once we find the horizontal component of the force, we can plug it into this formula, along with the mass, to find the acceleration.

Step-by-Step Solution: Cracking the Code

Now, let's get into the nitty-gritty and work through the problem step by step. First, we need to find the horizontal component of the force. Since the force is applied at an angle, we'll use trigonometry to break it down. Then we'll use Newton's Second Law of Motion, and finally, we'll arrive at the solution. I'll guide you through each part of the process, ensuring you fully grasp the concepts and calculations. This will help you understand the problem. It is really not that hard once you break it down into steps, so let's get started on calculating the horizontal acceleration.

Finding the Horizontal Component of Force

Alright, the first step is to calculate the horizontal component of the applied force. Because the force is applied at an angle, we need to find out how much of that force is actually pushing the box horizontally. To do this, we use the cosine function. The horizontal component (Fx) can be found using the formula: Fx = F * cos(θ). Here, F is the total force (100 N), and θ is the angle (36°). Plugging in the values, we get: Fx = 100 N * cos(36°). Using a calculator, cos(36°) is approximately 0.809. Therefore, Fx = 100 N * 0.809 = 80.9 N. This is the part of the force that's actually causing the box to move horizontally. Now, we have successfully found the horizontal component of the force.

Applying Newton's Second Law to Find Acceleration

Okay, now we've got the horizontal force (Fx = 80.9 N), and we know the mass of the box (m = 25 kg). We can now use Newton's Second Law (F = ma) to find the horizontal acceleration. First, we rearrange the formula to solve for acceleration: a = F / m. Plugging in our values, we get: a = 80.9 N / 25 kg. This gives us the acceleration. Calculating this, we find a = 3.236 m/s². This is the horizontal acceleration of the box. So, the horizontal acceleration of the box is approximately 3.24 m/s². This calculation uses the force we found in the previous step and the mass of the box to calculate its acceleration. This simple step leads us to our final answer.

The Answer and Explanation

Based on our calculations, the horizontal acceleration of the box is approximately 3.24 m/s². This corresponds to option B. So, the box accelerates horizontally at a rate of 3.24 meters per second squared when a 100 N force is applied at a 36° angle to a 25 kg box. The steps we took, breaking down the force into its components and using Newton's Second Law, are the standard approach for this kind of physics problem. Congratulations, we solved it! The key takeaways are understanding how to break down forces into components using trigonometry and how to apply Newton's Second Law.

Why the Other Options Are Incorrect

Let's briefly touch on why the other options aren't correct. Option A (1.52 m/s²) would be incorrect because it's a much lower acceleration, not accounting for the full horizontal component of the force. Option C (5.48 m/s²) is too high, likely resulting from an error in calculating the horizontal force component or in applying Newton's Second Law. Option D (6.87 m/s²) is even further off, and suggests a significant miscalculation, possibly due to a major misunderstanding of either the trigonometric or the Newton's Second Law formulas. So, the answer must be B. Only the correct answer matches up with the calculations, using the horizontal component of the force and Newton's Second Law.

Final Thoughts: Mastering Physics Problems

So there you have it, guys! We've successfully solved the physics problem and found the horizontal acceleration of the box. By breaking down the force into its components and using Newton's Second Law, we've demonstrated how to tackle these kinds of challenges. Remember that practice is key, so don't be afraid to try similar problems on your own. Keep in mind that understanding the fundamental concepts, such as force, mass, and acceleration, is crucial for success in physics. Keep practicing, and these concepts will become second nature! Remember to always apply the correct formula and follow your steps, and you will eventually succeed. Always take your time, and double-check your work.