Finding The Weight Of A In An Equilibrium System A Detailed Solution

Jul 12, 2025 by esunigon 69 views

Introduction

In the realm of physics, understanding equilibrium is fundamental. An object is said to be in equilibrium when the net force acting upon it is zero, resulting in no acceleration. This principle is crucial in various applications, from structural engineering to everyday mechanics. In this article, we will delve into a specific scenario involving an equilibrium system, where we aim to determine the weight of an object (A) given the weight of another object (B) and the condition that the system is in equilibrium. We will explore the underlying concepts, apply relevant formulas, and discuss the step-by-step solution to this problem. Grasping these concepts is not only essential for academic pursuits but also for understanding the forces at play in our physical world. So, let's embark on this journey to unravel the mysteries of equilibrium and weight determination.

Understanding Equilibrium

Equilibrium, in physics, is a state where the net force acting on an object is zero. This implies that the object is either at rest (static equilibrium) or moving with a constant velocity (dynamic equilibrium). Understanding equilibrium is crucial for solving problems related to forces and motion. When a system is in equilibrium, all the forces acting on it are balanced. This means that the sum of forces in any direction (horizontal and vertical) must be equal to zero. For instance, if we consider an object suspended by a rope, the tension in the rope must balance the weight of the object for it to remain in equilibrium. In more complex systems, such as those involving multiple forces acting at different angles, we need to resolve the forces into their components along the coordinate axes (usually x and y) and ensure that the sum of the components in each direction is zero. This principle is widely used in structural engineering, where ensuring equilibrium is paramount for the stability of buildings and bridges. In our specific problem, we are given that the system is in equilibrium, which is a critical piece of information that allows us to apply the principles of force balance to determine the unknown weight.

Problem Statement

Let's clearly state the problem at hand. We are tasked with finding the weight of object A, given that the weight of object B is 200N (Newtons). We also know that the entire system is in a state of equilibrium. This means that the forces acting on the system are balanced, and there is no net force causing acceleration. The options provided for the weight of A are:

A) 100N
B) 50N
C) 150N
D) 80N
E) 180N

To solve this problem effectively, we need to visualize the forces acting on the objects and apply the principles of equilibrium. This typically involves drawing a free-body diagram, which represents the object and all the forces acting on it. From the problem statement, it is evident that the key to finding the weight of A lies in understanding how the forces in the system balance each other. The fact that the system is in equilibrium provides us with a powerful tool—the ability to equate forces in opposing directions. This allows us to set up equations and solve for the unknown weight of A. The problem's simplicity is somewhat deceptive; it hinges on a clear understanding of equilibrium and the ability to apply this concept to a practical situation.

Setting up the Equations

To solve for the weight of object A, we need to translate the problem's conditions into mathematical equations. Since the system is in equilibrium, we know that the forces acting on it are balanced. This is a crucial concept, and it's the foundation of our solution.

First, visualize the system. Imagine object A and object B connected in some way, possibly by ropes or cables, and suspended in such a manner that the entire system is not moving. The weight of object B (200N) acts downwards. To maintain equilibrium, there must be an equal and opposite force acting upwards. This upward force is likely provided by the tension in the supporting cables or ropes.

Now, let's denote the weight of object A as ${W_A}$ . The weight of object B is given as ${W_B = 200N}$ . If the system involves a simple setup where the weights are directly balancing each other (for example, over a pulley), then the weight of A must be directly related to the weight of B. However, without a specific diagram or more details about the configuration of the system, we must make some assumptions. If we assume the simplest scenario where the weights are balanced vertically, we can say that the total upward force must equal the total downward force.

Mathematically, this can be represented as: ${W_A = W_B}$ if they are directly balancing each other. However, the setup might involve angles or other complexities, which would require resolving forces into components. For a more complex scenario, we would need to consider the angles at which the forces are acting and use trigonometric functions to find the components of the forces in the vertical and horizontal directions. In such cases, the equilibrium conditions would be expressed as the sum of forces in the x-direction equaling zero and the sum of forces in the y-direction equaling zero.

Without a diagram, we will proceed with the simplest assumption for now and adjust our approach if needed later. Based on the direct balancing assumption, we can set up our equation as follows: ${W_A = 200N}$ . However, this initial setup seems too straightforward given the provided options. This suggests that there might be a more intricate arrangement where the weight of A is not simply equal to the weight of B. We'll revisit this assumption and consider other possibilities as we proceed with the analysis.

Solving for the Weight of A

Given our initial setup and the problem's context, it's clear that we need to refine our approach. The direct equality of weights ( ${W_A = W_B}$ ) doesn't align with the answer choices, indicating a more complex scenario. We need to consider a situation where the weights are not directly balanced but are part of a system involving tension and possibly angles.

Let's assume a scenario where objects A and B are suspended by ropes that meet at a point, creating a configuration where the tensions in the ropes support the weights. In this case, the tensions in the ropes would have vertical components that balance the weights of A and B. The horizontal components of the tensions would also need to balance each other to maintain equilibrium.

Without a specific diagram, we'll make a further assumption to proceed with the calculation. Let's assume the ropes holding A and B make specific angles such that the tension supporting A is related to the tension supporting B. A common scenario in introductory physics problems is one where the angles allow for simple trigonometric relationships. For instance, if the ropes form a symmetrical arrangement, the tensions might be related by factors involving sines or cosines of certain angles.

To illustrate, let's hypothetically consider a case where the rope supporting A makes an angle such that its vertical component of tension is half the tension supporting B. This is a simplified assumption for demonstration purposes. In a real problem, we would need specific angles to perform accurate calculations.

If we let ${T_A}$ be the tension in the rope supporting A and ${T_B}$ be the tension in the rope supporting B, and assume that the vertical component of ${T_A}$ that supports A is related to ${T_B}$ by some factor, we can write:

${T_A \cdot \sin(\theta) = W_A}$

${T_B \cdot \sin(\phi) = W_B}$

Where ${\theta}$ and ${\phi}$ are the angles the ropes make with the horizontal or vertical, and ${\sin(\theta)}$ and ${\sin(\phi)}$ represent the vertical components of the tensions.

If we assume a specific relationship, say ${T_A = \frac{1}{2} T_B}$ (again, this is a hypothetical scenario without a diagram), and if we knew the angles, we could solve for ${W_A}$ . However, without more information, we must try a different approach based on the answer choices.

Given the options, we can test which weight for A would make sense in an equilibrium system with B weighing 200N. If we consider option A (100N), this suggests that A might be supported by a tension force that is half of the weight of B. This aligns with our hypothetical scenario where the vertical component of tension supporting A is less than that supporting B. Therefore, 100N is a plausible answer.

If we consider option C (150N), this would imply a more complex relationship between the tensions and angles, which is less likely given the typical structure of introductory physics problems. The same logic applies to options D (80N) and E (180N).

Therefore, based on our analysis and the given options, option A (100N) appears to be the most logical solution. We acknowledge that this solution is based on assumptions due to the lack of a specific diagram, and a precise answer would require more information about the system's configuration.

Analyzing the Options

In this section, we will meticulously analyze the given options to determine the most plausible answer for the weight of object A. Recall that the weight of object B is 200N, and the system is in equilibrium. We have the following options for the weight of A:

A) 100N
B) 50N
C) 150N
D) 80N
E) 180N

When dealing with equilibrium problems, it's essential to consider how the forces balance each other. The weight of an object is the force exerted on it due to gravity, and in an equilibrium system, this force must be counteracted by other forces, such as tension in ropes or normal forces from surfaces. Let's evaluate each option in the context of equilibrium.

Option A) 100N: If the weight of A is 100N, this implies that the force needed to support A is half the force needed to support B (which weighs 200N). This is plausible in scenarios where the objects are suspended by ropes at angles. For instance, if the rope supporting A is at an angle, the vertical component of the tension in the rope would be less than the total tension, allowing a smaller weight for A to be balanced. This option suggests a balanced system where the forces are proportional, which aligns with common equilibrium scenarios.
Option B) 50N: If the weight of A is 50N, this would mean the force supporting A is significantly less than the force supporting B. While this is possible, it would likely require a specific configuration, such as a pulley system or angled supports, where the mechanical advantage reduces the force needed to support A. This option is less straightforward but not entirely improbable.
Option C) 150N: If the weight of A is 150N, the forces supporting A and B would be closer in magnitude. This could occur in a system where the supports are more direct, without significant mechanical advantage or angled forces. However, it's less intuitive than option A, which offers a simpler proportional relationship.
Option D) 80N: A weight of 80N for A suggests a similar scenario to option B, where the support system provides some mechanical advantage or angled forces. This option is also plausible but less likely than option A due to the slightly more complex force relationships it implies.
Option E) 180N: If the weight of A is 180N, this is close to the weight of B (200N). This scenario would imply a very direct support system with minimal mechanical advantage or angled forces. While possible, it's less likely given the typical problem structure in introductory physics, where simpler proportional relationships are often the key.

Considering the analysis, option A (100N) appears to be the most logical choice. It presents a balanced scenario where the forces can be easily related, and it aligns with common equilibrium problem setups. The other options are plausible but less straightforward, requiring more specific and complex system configurations.

Conclusion

In conclusion, the problem of finding the weight of object A in an equilibrium system highlights the importance of understanding force balance and applying the principles of equilibrium. Given that the weight of object B is 200N and the system is in equilibrium, we explored various scenarios and analyzed the provided options. Our analysis indicated that option A, 100N, is the most plausible answer. This conclusion is based on the assumption of a balanced system where the forces are proportional, a common scenario in introductory physics problems.

However, it is crucial to acknowledge that the precise solution would depend on a detailed understanding of the system's configuration, including the angles of suspension, the presence of pulleys, and other factors that influence the force distribution. Without a specific diagram or additional information, we made logical assumptions to arrive at the most reasonable answer. This underscores the significance of visualizing the forces acting on the objects and setting up appropriate equations based on the equilibrium conditions.

By systematically evaluating the options and applying the principles of equilibrium, we demonstrated a method for solving this type of problem. While our solution is based on certain assumptions, it provides a framework for approaching similar problems in the future. Remember, in physics, a clear understanding of the underlying concepts and a methodical approach are essential for problem-solving. This exercise not only reinforces the concepts of equilibrium but also highlights the importance of careful analysis and logical reasoning in tackling physics problems.

Therefore, based on our analysis, the most likely weight of object A is 100N.