Problem 8 - Calculus of a Single Variable

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. In order to move a 600 -pound rock, a person weighing 200 pounds wants to balance it on a beam that is 5 feet long.

Step-by-Step Solution

Answer

The person will have to position the fulcrum 1.67 feet from the end of the beam on which the rock is placed to balance the system.

Step 1: Identify given values

The weight of the rock \(W_{1} = 600\) pounds, the weight of the person \(W_{2} = 200\) pounds, and the length of the beam \(L = 5\) feet.

Step 2: Set up the equilibrium equation

Taking moments about the point where the person is standing since we know both the force (her weight \(W_{2}\)) and its distance from the point (length of the board \(L\)). This gives the equation \(W_{1} \cdot x = W_{2} \cdot L\) to solve for \(x\) .

Step 3: Complete calculations

Substitute the given values into the equilibrium equation and solve for \(x\). This gives \(x = \frac{{W_{2} \cdot L}}{{W_{1}}}\). Substituting the given values gives \(x = \frac{{200 \cdot 5}}{{600}}\) which simplifies to \(x = 1.67\) feet.

Key Concepts

Fulcrum

Imagine balancing a pencil on your finger. The point where the pencil touches your finger and doesn't fall off is known as the fulcrum. In physics, this concept is key to understanding how seesaws, levers, and even crowbars work. A fulcrum is the pivot point around which a lever rotates, and it allows us to lift heavy objects with less effort.

Referring back to our original problem, the fulcrum is the point on the beam where it is supported and around which it can pivot. When objects of different weights are placed at varying distances from the fulcrum, it's the task of physics to help us locate the fulcrum precisely to achieve balance, or mechanical equilibrium. For the person trying to move the rock, finding the perfect spot for the fulcrum means they can move the heavy object with ease.

Moments

A moment is a measure of the force causing an object to rotate around a point, and this point is often the fulcrum. The moment is calculated by multiplying the force applied (its weight, in the case of gravity) by the distance from the fulcrum. Moments have a direction; counterclockwise moments and clockwise moments must be equal for an object to be in equilibrium.

Heavier objects don't always create larger moments. The distance from the fulcrum is crucial. In our exercise, we have two objects with different weights placed on opposite ends of a beam. To achieve balance, the moments caused by each weight around the fulcrum must be equal, essentially creating a situation where the product of each weight and its respective distance from the fulcrum are equal.

Equilibrium Equations

When we are seeking a state where there are no net forces or moments causing rotation, we are talking about equilibrium. Equilibrium equations are the math behind this concept, providing a simple way to calculate the conditions needed for balance. In mechanical systems, there are typically two types of equilibrium: translational, where forces are balanced so that there's no linear movement, and rotational, where moments are balanced to prevent rotation.

In the context of our problem, we're concerned with rotational equilibrium. By setting the counterclockwise (caused by the person's weight) and clockwise (caused by the rock's weight) moments equal to each other, we derive our equilibrium equation: \(W_1 \cdot x = W_2 \cdot L\). This equation is a golden rule for ensuring that our beam, with weights on either end, doesn't rotate and stays perfectly balanced.