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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem: Balancing a Beam with Weights**
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.
In order to move a 600-pound rock, a person weighing 100 pounds wants to balance it on a beam that is 5 feet long. Find \( x \) such that the system is in equilibrium.
![Diagram Description]
The diagram consists of a beam balanced on a fulcrum. Weight \( W_1 \) (100 pounds) is placed on one end of the beam and weight \( W_2 \) (600 pounds) on the other end. The distances from the fulcrum are noted as \( x \) and \( L - x \) respectively.
### Step 1: Assign the Values
\[
W_2 = 600
\]
\[
W_1 = 100
\]
\[
L = 5 \text{ ft}
\]
### Step 2: Determine the Distance from the Fulcrum
Assume the person has to sit \( x \) feet away from the fulcrum in order to balance the system. Assume the fulcrum is located at the origin. Therefore, if \( x_2 = x \), then:
\[
x_1 = L - x
\]
\[
x_1 = 5 - x
\]
### Step 3: Ensure Equilibrium
For equilibrium, the resultant moment should be 0.
\[
M_0 = 0 = W_1 x_1 + W_2 x_2
\]
Therefore,
\[
0 = 100 (5 - x) - 600 x
\]
Simplify:
\[
100 (5 - x) = 600 x
\]
\[
500 - 100x = 600x
\]
\[
500 = 700x
\]
\[
x = \frac{500}{700}
\]
\[
x = \frac{5}{7} \text{ ft}
\]
Thus, the person must sit \( \frac{5}{7} \) feet away from the fulcrum to balance the system.
---
### Explanation of Diagram
The diagram shows a horizontal beam (5 feet in length) balanced on](https://content.bartleby.com/qna-images/question/0fdef185-de8b-42f5-933e-8018dcf36541/2d4e76dd-8921-4c7f-814d-243a1ca76189/060692k_thumbnail.png)
Transcribed Image Text:**Problem: Balancing a Beam with Weights**
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.
In order to move a 600-pound rock, a person weighing 100 pounds wants to balance it on a beam that is 5 feet long. Find \( x \) such that the system is in equilibrium.
![Diagram Description]
The diagram consists of a beam balanced on a fulcrum. Weight \( W_1 \) (100 pounds) is placed on one end of the beam and weight \( W_2 \) (600 pounds) on the other end. The distances from the fulcrum are noted as \( x \) and \( L - x \) respectively.
### Step 1: Assign the Values
\[
W_2 = 600
\]
\[
W_1 = 100
\]
\[
L = 5 \text{ ft}
\]
### Step 2: Determine the Distance from the Fulcrum
Assume the person has to sit \( x \) feet away from the fulcrum in order to balance the system. Assume the fulcrum is located at the origin. Therefore, if \( x_2 = x \), then:
\[
x_1 = L - x
\]
\[
x_1 = 5 - x
\]
### Step 3: Ensure Equilibrium
For equilibrium, the resultant moment should be 0.
\[
M_0 = 0 = W_1 x_1 + W_2 x_2
\]
Therefore,
\[
0 = 100 (5 - x) - 600 x
\]
Simplify:
\[
100 (5 - x) = 600 x
\]
\[
500 - 100x = 600x
\]
\[
500 = 700x
\]
\[
x = \frac{500}{700}
\]
\[
x = \frac{5}{7} \text{ ft}
\]
Thus, the person must sit \( \frac{5}{7} \) feet away from the fulcrum to balance the system.
---
### Explanation of Diagram
The diagram shows a horizontal beam (5 feet in length) balanced on
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