Calculating Work: Step-by-Step Guide

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Calculating Work: Step-by-Step Guide

Hey guys! Let's dive into a physics problem that's super common: calculating work. We'll be using the formula τ = f ⋅ d ⋅ cosθ, where f represents force, d represents displacement, and θ is the angle between the force and the displacement. Specifically, we're given that f = 40 N (Newtons), d = 20 m (meters), and θ = 135°. Don't worry, it sounds more complicated than it is! I'll walk you through each step and make it as easy as possible. This is a fundamental concept in physics, and understanding it can unlock a whole world of problem-solving. We'll break down the formula, the values, and the calculations to find the final result, ensuring you grasp the concept of work done by a force.

Understanding the Work Formula

Alright, first things first: let's get friendly with the formula. The formula τ = f ⋅ d ⋅ cosθ tells us how to calculate work (τ). Think of work as the energy transferred by a force when it causes an object to move. It's not just about pushing or pulling; it's about the effect of that push or pull over a distance. The formula takes into account three key things. First is the force (f). This is how hard you're pushing or pulling, measured in Newtons (N). Second, we have the displacement (d), which is the distance the object moves. This is measured in meters (m). Finally, we have the angle (θ) between the force and the direction the object moves. The angle is super important because it determines how much of the force is actually doing the work. If you're pushing at an angle, only part of your push is effectively moving the object forward. The cosθ part of the formula accounts for this. If the force and displacement are in the same direction, the angle is 0°, and cos(0°) = 1, meaning all the force is contributing to the work. If the force is perpendicular to the displacement (like when you're carrying a box horizontally), the angle is 90°, and cos(90°) = 0, meaning no work is done. Make sense, right? This formula is the cornerstone of understanding how forces interact with objects to cause movement and energy transfer, so take your time to fully understand the components.

Now, let's look at the units. Force is measured in Newtons (N), distance in meters (m), and work, the result, is measured in Joules (J). One Joule is the same as one Newton-meter (Nâ‹…m). So, if you see those units, you're on the right track!

Step-by-Step Calculation: Finding the Work

Okay, time for the fun part: plugging in the numbers! We've got the formula, and we've got the values. Here's how we'll break it down:

  1. Identify the values: We know that f = 40 N, d = 20 m, and θ = 135°.

  2. Plug the values into the formula: We'll substitute these values into our equation: τ = 40 N ⋅ 20 m ⋅ cos(135°). Make sure you're using the correct units! It's super easy to mess up if you don't keep track of those.

  3. Calculate the cosine: Use a calculator to find the cosine of 135°. cos(135°) ≈ -0.7071. Remember that the cosine of an angle greater than 90° will be negative. This is because the force has a component that is opposing the direction of motion.

  4. Multiply: Now, multiply all the values together: τ = 40 N ⋅ 20 m ⋅ (-0.7071). Doing the math gives us τ ≈ -565.68 J.

So there you have it! The work done by the force is approximately -565.68 Joules. The negative sign is crucial and indicates that the force is acting in a direction opposite to the displacement. This could mean the force is resisting the motion, like friction. The steps are pretty straightforward, right? Identify, substitute, calculate, and remember those units! It's all about systematically breaking down the problem.

Interpreting the Result

What does -565.68 Joules actually mean? The negative sign is super important! It tells us that the work done is negative. In this specific scenario, this negative sign indicates that the force is working against the displacement. For instance, the angle being at 135° means the force is partially opposing the object's movement. It could mean the force is friction, opposing the object's motion.

If the angle was between 0° and 90°, the result would have been positive, which would mean that the force is contributing to the object's motion. If the angle was exactly 90°, the cosine is zero, which would mean the work done is zero. This happens, for example, when you are carrying a box horizontally. You are applying force to the box, but because there is no displacement in the direction of the force, no work is done by your arms on the box. It’s all about the interplay between force, displacement, and the angle between them. So, the negative sign here indicates that the force is, in a sense, taking away energy from the object’s motion, or at least opposing it.

Choosing the Correct Alternative

Let's assume the multiple-choice options were something like this:

a. -565.68 J b. 565.68 J c. 0 J d. 800 J

Based on our calculations and the explanation, the correct answer would be a. -565.68 J. Remember, pay close attention to the sign (positive or negative) and the units (Joules). The other options are incorrect because they either have the wrong sign or the wrong numerical value, which results from a calculation error or misunderstanding of the physics concepts involved. Always double-check your work and make sure you're using the correct formula and units to avoid any confusion and get the right answer.

Conclusion: Mastering Work Calculations

There you have it! Calculating work using the formula τ = f ⋅ d ⋅ cosθ. We've covered the formula, the steps, the calculation, and, most importantly, the meaning of the result. Remember, understanding the components of the formula (force, displacement, and the angle) is key. Take your time, break down the problem step-by-step, and always keep track of the units. Practicing with different values and angles will help you master this concept, so go ahead and try some similar problems. The more you work with it, the easier it will become. Keep in mind that physics is all about understanding how the world around us works, and this formula is a powerful tool to understand the relationship between force, displacement, and energy transfer. Keep up the awesome work, and happy calculating!