Tension Calculation: Equilibrium Of A 20 N Weight

by Dimemap Team 50 views

Hey guys! Let's dive into a physics problem where we need to figure out the tension in two strings holding a weight. This is a classic problem in statics, and understanding it will help you grasp the concepts of forces, equilibrium, and trigonometry in physics.

Problem Setup: Weight and Strings

Let’s break down the problem. We have a weight of 20 N (Newtons, the unit of force) hanging in the air. This weight is held in place, or in equilibrium, by two strings, which we’ll call F1 and F2. Now, here’s the interesting part: both of these strings are at a 45° angle with the vertical. This angle is crucial because it allows us to use trigonometry to figure out the forces involved. We’re also given that sin(45°) = cos(45°) = √2/2, which will simplify our calculations later. So, the main question here is: what is the tension in each of these strings? Tension, in this context, refers to the pulling force exerted by each string. Understanding how to calculate tension is fundamental in many areas of physics and engineering, from designing bridges to analyzing simple machines. So, let's get started!

Understanding Forces and Equilibrium

Before we jump into the math, it’s super important to understand the physics concepts at play here. The first key concept is force. Force is simply a push or a pull. In this problem, we have three forces acting: the weight of the object pulling downwards due to gravity, and the tension forces F1 and F2 pulling upwards and outwards. The second key concept is equilibrium. Equilibrium means that the object is not moving; it's perfectly still. For an object to be in equilibrium, all the forces acting on it must balance out. This means that the sum of all forces in any direction must be zero. Think of it like a tug-of-war where both sides are pulling with equal force – the rope doesn't move because the forces are balanced. In our problem, the upward forces from the strings must balance the downward force of the weight. To analyze these forces effectively, we'll need to break them down into their horizontal and vertical components. This is where our knowledge of trigonometry, particularly sine and cosine, comes into play. By understanding these fundamental principles, we can tackle this problem with confidence and apply these concepts to other similar scenarios in mechanics.

Breaking Down Forces into Components

The core of solving this problem lies in breaking down the tension forces (F1 and F2) into their horizontal and vertical components. Why do we do this? Because it allows us to analyze the forces acting in each direction independently. Imagine each tension force as a vector, which is like an arrow with a certain length (magnitude) and direction. We can think of this vector as having two 'shadows': one on the horizontal axis and one on the vertical axis. These shadows are the horizontal and vertical components of the force. To find these components, we use trigonometry. Since the strings are at a 45° angle to the vertical, we can use sine and cosine to calculate the components. The vertical component of each tension force (F1y and F2y) will be equal to the magnitude of the tension force multiplied by the cosine of the angle (cos(45°)), while the horizontal component (F1x and F2x) will be equal to the magnitude of the tension force multiplied by the sine of the angle (sin(45°)). Remember, we’re given that sin(45°) = cos(45°) = √2/2, which makes our calculations a bit easier. Breaking down the forces like this allows us to see clearly how the upward forces from the strings counteract the downward force of the weight, and how the horizontal forces balance each other out to maintain equilibrium. This step is crucial for setting up the equations we'll use to solve for the tension in each string.

Calculating Tension in Each String

Now, let's get to the heart of the problem: calculating the tension in strings F1 and F2. We know the weight being supported is 20 N, and this force acts vertically downwards. For the weight to be in equilibrium, the total upward force provided by the vertical components of the tension in the strings must equal the weight. Mathematically, this can be expressed as: F1y + F2y = 20 N. Remember, F1y and F2y are the vertical components of the tension forces F1 and F2, respectively. Since both strings are at the same angle (45°) to the vertical and we're assuming the system is symmetrical, the tension in both strings will be equal (F1 = F2). This simplifies our equation significantly. Let's call the tension in each string T. Then, F1y = T * cos(45°) and F2y = T * cos(45°). Substituting these into our equilibrium equation, we get: T * cos(45°) + T * cos(45°) = 20 N. We know that cos(45°) = √2/2, so we can plug that in: T * (√2/2) + T * (√2/2) = 20 N. Now, it’s just a matter of solving for T. By simplifying and isolating T, we can find the magnitude of the tension force in each string. This step-by-step approach ensures that we're not just plugging numbers into a formula, but actually understanding the physics behind the calculations.

Step-by-Step Solution

Alright, guys, let's walk through the solution step-by-step to make sure we’ve got this down.

  1. We start with the equilibrium equation: T * (√2/2) + T * (√2/2) = 20 N.
  2. Combine the terms on the left side: 2 * T * (√2/2) = 20 N, which simplifies to T * √2 = 20 N.
  3. Now, we want to isolate T, so we divide both sides of the equation by √2: T = 20 N / √2.
  4. To rationalize the denominator (get rid of the square root in the denominator), we multiply both the numerator and the denominator by √2: T = (20 N * √2) / (√2 * √2).
  5. This simplifies to T = (20 N * √2) / 2.
  6. Finally, we simplify further by dividing 20 by 2: T = 10√2 N.

So, the tension in each string is 10√2 N. This is the magnitude of the force that each string is exerting to hold the weight in equilibrium. If you want a decimal approximation, you can use a calculator to find that √2 is approximately 1.414, so T is approximately 14.14 N. Breaking down the problem into these clear steps makes it easier to follow and understand the logic behind each calculation. Now you've got it!

Horizontal Forces and Equilibrium

While we focused on the vertical forces to calculate the tension, it’s important not to forget about the horizontal forces acting in this system. In our setup, the horizontal components of the tension forces, F1x and F2x, are also crucial for maintaining equilibrium. However, they don't directly contribute to supporting the weight. Instead, they balance each other out. Remember, for the system to be in equilibrium, the sum of forces in both the vertical and horizontal directions must be zero. So, we have F1x acting in one direction and F2x acting in the opposite direction. Because the angles and tensions are symmetrical in this problem (both strings are at 45° and have equal tension), the magnitudes of F1x and F2x will be the same, but their directions are opposite. This means they perfectly cancel each other out, ensuring there's no net horizontal force on the weight. This might seem like a minor point, but it’s a fundamental aspect of static equilibrium. Understanding that all forces, in all directions, must balance is key to solving more complex problems involving forces and structures. In more complicated scenarios where the angles or tensions are different, you’d need to carefully calculate each horizontal component to ensure they balance. Ignoring these horizontal forces can lead to incorrect conclusions about the overall stability and equilibrium of a system. Always consider all directions!

Conclusion: Tension Achieved!

Alright, guys, we’ve successfully calculated the tension in each string! We started with a problem involving a 20 N weight suspended by two strings at a 45° angle, and we systematically broke it down using the principles of physics and trigonometry. We found that the tension in each string is 10√2 N, which is approximately 14.14 N. This problem highlights the importance of understanding concepts like force, equilibrium, and how to resolve forces into components. By breaking down the forces into their vertical and horizontal components, we could apply the conditions for equilibrium and solve for the unknowns. Remember, the key to solving physics problems is not just plugging in numbers, but understanding the underlying principles and applying them logically. This example can serve as a foundation for tackling more complex problems involving forces and tensions in various scenarios. Keep practicing, and you'll become a force to be reckoned with in physics!