Problem 7
Question
Consider a beam of length \(L\) with a fulcrum \(x\) feet from one end (see figure). There are objects with weights \(W_{1}\) and \(W_{2}\) placed on opposite ends of the beam. Find \(x\) such that the system is in equilibrium. Two children weighing 48 pounds and 72 pounds are going to play on a seesaw that is 10 feet long.
Step-by-Step Solution
Verified Answer
The fulcrum should be 4 feet from the child weighing 72 pounds.
1Step 1: Set up the formula
First arrange the principle of moments formula: \(W_{1} * (L - x) = W_{2} * x\), where \(W_{1}\) and \(W_{2}\) are the weights on either side of the fulcrum, \(L\) is the total length of the seesaw, and \(x\) is the distance we are trying to find.
2Step 2: Substitute known values
Plug in the given values into the equation: \(48 * (10 - x) = 72 * x\). Now the formula is ready for solving.
3Step 3: Simplify and solve for \(x\)
Simplify the equation, which results in distributing \(48\) into the brackets and multiplying \(72\) and \(x\) together. We have \(480 - 48x = 72x\). Rearrange terms to have \(x\) on one side of the equation: \(480 = 120x\). So, solving for \(x\), we get \(x = 4\).
Key Concepts
Principle of MomentsLever EquilibriumSolving for Unknowns in Physics
Principle of Moments
The principle of moments, also known as the lever rule, is a fundamental concept in physics concerning the balance of forces and the turning effect produced by them — the moments — around a pivot point or fulcrum. In essence, it states that for a lever to be in equilibrium, the sum of the clockwise moments about the pivot must equal the sum of the counterclockwise moments.
Understanding the Principle: Imagine a simple seesaw with two weights at its ends. When the seesaw is balanced, the product of the weight on one side and its distance from the fulcrum is equal to the product on the other side. Formally, this is expressed as:\[W_{1} \times d_{1} = W_{2} \times d_{2}\], where \(W_{1}\) and \(W_{2}\) are the weights and \(d_{1}\) and \(d_{2}\) are the respective distances from the fulcrum. To achieve equilibrium, you can adjust either the weights or the distances.
When applying the principle to real-world problems, factors like the distribution of weight and the presence of multiple forces need to be considered. However, the core principle remains: balance the moments for stability.
Understanding the Principle: Imagine a simple seesaw with two weights at its ends. When the seesaw is balanced, the product of the weight on one side and its distance from the fulcrum is equal to the product on the other side. Formally, this is expressed as:\[W_{1} \times d_{1} = W_{2} \times d_{2}\], where \(W_{1}\) and \(W_{2}\) are the weights and \(d_{1}\) and \(d_{2}\) are the respective distances from the fulcrum. To achieve equilibrium, you can adjust either the weights or the distances.
When applying the principle to real-world problems, factors like the distribution of weight and the presence of multiple forces need to be considered. However, the core principle remains: balance the moments for stability.
Lever Equilibrium
Lever equilibrium occurs when a lever system is balanced, with no net moment causing it to rotate. This is closely related to the principle of moments, and it is essential in understanding how to manipulate a lever system to achieve a balance.
Real-life Application: Using the seesaw example, the challenge is to ascertain where to place the fulcrum or how to distribute the weights to attain equilibrium. In the exercise, we have two children with different weights aiming to balance on a seesaw. If their weights are not identical, the fulcrum needs to be positioned strategically. It's not just about the force exerted by the weights but also how far these forces are from the pivot point.
By adjusting the position of the fulcrum (or in the case of the exercise, solving for \(x\)), equilibrium can be achieved even with different magnitudes of weight. The goal is to make the seesaw level by ensuring the product of the weight and distance from the fulcrum is uniform across both sides.
Real-life Application: Using the seesaw example, the challenge is to ascertain where to place the fulcrum or how to distribute the weights to attain equilibrium. In the exercise, we have two children with different weights aiming to balance on a seesaw. If their weights are not identical, the fulcrum needs to be positioned strategically. It's not just about the force exerted by the weights but also how far these forces are from the pivot point.
By adjusting the position of the fulcrum (or in the case of the exercise, solving for \(x\)), equilibrium can be achieved even with different magnitudes of weight. The goal is to make the seesaw level by ensuring the product of the weight and distance from the fulcrum is uniform across both sides.
Solving for Unknowns in Physics
Physics problems often involve solving for unknowns — values which are not immediately evident but are crucial for understanding a system's behavior. Whether it's finding the distance, time, force, or, as in our example, the position of a fulcrum, the ability to manipulate equations and apply physical laws is key.
Step-by-Step Approach: To solve for an unknown, start by identifying the relevant principles and formulas that apply to the problem. In the given exercise, we initially establish the principle of moments formula:\[ W_{1} \times (L - x) = W_{2} \times x\]. The next step involves substituting the known values into this formula. Finally, by algebraic manipulation, we isolate the unknown variable and solve for it. In the process, simplify where possible to make the calculations easier.
This methodical approach turns complex problems into more manageable steps, ultimately revealing the unknowns and deepening the understanding of the physical concepts at play.
Step-by-Step Approach: To solve for an unknown, start by identifying the relevant principles and formulas that apply to the problem. In the given exercise, we initially establish the principle of moments formula:\[ W_{1} \times (L - x) = W_{2} \times x\]. The next step involves substituting the known values into this formula. Finally, by algebraic manipulation, we isolate the unknown variable and solve for it. In the process, simplify where possible to make the calculations easier.
This methodical approach turns complex problems into more manageable steps, ultimately revealing the unknowns and deepening the understanding of the physical concepts at play.
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