Problem 1 - Thomas Calculus

Question

An \(80-\) lb child and a \(100-\) lb child are balancing on a seesaw. The \(80-\) lb child is 5 \(\mathrm{ft}\) from the fulcrum. How far from the fulcrum is the \(100-\) lb child?

Step-by-Step Solution

Answer

The 100-lb child should sit 4 feet from the fulcrum.

Step 1: Understand the Problem

We need to find out how far the 100-lb child should sit from the fulcrum on a seesaw to keep it balanced, given that the 80-lb child is 5 feet from the fulcrum.

Step 2: Seesaw Balance Principle

A seesaw will be balanced when the product of the weight of each child and their distance from the fulcrum are equal on both sides. This means that the torque (moment of force) about the fulcrum must be equal for both children.

Step 3: Set Up the Equation

Let the distance of the 100-lb child from the fulcrum be denoted by \( x \). To balance the seesaw, the torque produced by the 80-lb child should equal the torque produced by the 100-lb child. The equation is:\[ 80 \text{ lb} \times 5 \text{ ft} = 100 \text{ lb} \times x \text{ ft} \]

Step 4: Solve the Equation

Simplify the equation from the previous step to solve for \( x \):\[ 400 = 100x \]\[ x = \frac{400}{100} \]\[ x = 4 \]

Step 5: Conclusion

The 100-lb child should sit 4 feet from the fulcrum to balance the seesaw.

Key Concepts

Understanding Torque in Physics

In physics, torque is a measure of the rotational force acting on an object. Imagine pushing a door open; the distance from the hinge (fulcrum) to where you apply the force matters. The seesaw balance problem demonstrates this concept perfectly.
Torque is calculated by multiplying the force applied by the distance from the pivot point or fulcrum as in a lever system. In our problem:

• Torque for the 80-lb child: \(80 \times 5\). This gives us a torque of 400 lb-ft.
• Conversely, the 100-lb child's torque depends on their distance, \(x\), from the fulcrum. The equation is \(100 \times x\).

The goal is to equalize the torques, ensuring that both sides of the seesaw exert the same rotational force about the fulcrum.

Balancing and Equilibrium

Equilibrium occurs when all the forces or moments acting on a system result in a stable situation, with no resultant force or rotation. In our seesaw problem, achieving equilibrium implies that the seesaw is perfectly balanced and neither side is tilting.
For a seesaw, balance is achieved when the product of the weight and distance from the fulcrum is equal on both sides, meaning both children must apply equal and opposite torques.

• This ensures that the rotational forces cancel each other out, preventing any movement.
• In a balanced state, the net torque is zero, indicating perfect equilibrium.

Approaches to Solving Physics Problems

Physics problem-solving often involves strategic steps to break down complex problems like the seesaw balance problem into manageable parts. Here's how we can approach similar problems:

• Identify known and unknown variables. For our problem, the knowns are the weights and the 80-lb child’s distance. The unknown is the distance \(x\) for the 100-lb child.
• Understand the underlying principles, such as the seesaw equilibrium in our case.
• Develop a mathematical model with equations reflecting the physics laws, like balancing torques here.
• Solve the algebraic equation to find the unknown variable, then verify by checking units and logical consistency.

This methodical approach is crucial for dissecting any physics-related issues.

Using Algebraic Equations to Resolve Torque Problems

Algebra serves as an invaluable tool for solving physics problems by allowing us to create equations based on known principles.
In our example:

• We have an equation \(80 \times 5 = 100 \times x\) arising from setting equal the torques of the two children on the seesaw.
• Here, arithmetic operations simplify the equation to \(400 = 100x\).
• To find \(x\), divide each side by 100, resulting in \(x = 4\).

Solving this algebraic equation provides us the exact distance ensuring equilibrium. This real-world application of algebra demonstrates how mathematical skills directly solve physical phenomena, making it less daunting and more intuitive.