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A homeowner is trying to move a stubborn rock from his yard. By using a a metal rod as a lever arm and a fulcrum (or pivot point) the homeowner will have a better chance of moving the rock. The homeowner places the fulcrum a distance ?=0.266 m from the rock, which has a mass of 325 kg, and fits one end of the rod under the rock's center of weight.
![### Problem Overview:
A homeowner is attempting to move a large rock in their yard using a lever system. By employing a metal rod as a lever and utilizing a fulcrum (pivot point), the homeowner can enhance their ability to move the rock. The fulcrum is positioned \( d = 0.266 \, \text{m} \) away from the rock, which has a mass of 325 kg, and the rod is placed beneath the rock's center of weight.
### Objective:
The task is to determine the minimum total length \( L \) of the rod required to move the rock when the homeowner applies a maximum force of 663 N at the rod's opposite end. Assume the rod is massless and oriented nearly horizontally, ensuring the weight of the rock and the homeowner's force remain essentially vertical. The acceleration due to gravity is \( g = 9.81 \, \text{m/s}^2 \).
### Diagram Explanation:
The diagram illustrates a large rock beside a lever system. A rod is shown operating as a lever, with one end placed beneath the rock and a fulcrum located a distance \( d \) from the rock. The rod extends beyond the fulcrum, where the homeowner applies force.
### Formula Application:
To solve for the minimum length \( L \) of the rod, use the principle of torque balance around the fulcrum:
\[
\text{Torque by rock} = \text{Torque by homeowner's force}
\]
\[
m \cdot g \cdot d = F \cdot (L - d)
\]
Where:
- \( m = 325 \, \text{kg} \) is the mass of the rock
- \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity
- \( d = 0.266 \, \text{m} \) is the distance from the fulcrum to the rock
- \( F = 663 \, \text{N} \) is the force applied by the homeowner
### Solution:
Solve the equation for \( L \):
\[
325 \times 9.81 \times 0.266 = 663 \times (L - 0.266)
\]
Determine \( L \) to ensure the lever system efficiently moves the rock.](https://content.bartleby.com/qna-images/question/f90e9f76-a8ee-4e86-857e-4cb9f0bb2381/c4bde972-4e1f-4eca-9adf-679af20c06e5/142ai88_processed.png)
Transcribed Image Text:### Problem Overview:
A homeowner is attempting to move a large rock in their yard using a lever system. By employing a metal rod as a lever and utilizing a fulcrum (pivot point), the homeowner can enhance their ability to move the rock. The fulcrum is positioned \( d = 0.266 \, \text{m} \) away from the rock, which has a mass of 325 kg, and the rod is placed beneath the rock's center of weight.
### Objective:
The task is to determine the minimum total length \( L \) of the rod required to move the rock when the homeowner applies a maximum force of 663 N at the rod's opposite end. Assume the rod is massless and oriented nearly horizontally, ensuring the weight of the rock and the homeowner's force remain essentially vertical. The acceleration due to gravity is \( g = 9.81 \, \text{m/s}^2 \).
### Diagram Explanation:
The diagram illustrates a large rock beside a lever system. A rod is shown operating as a lever, with one end placed beneath the rock and a fulcrum located a distance \( d \) from the rock. The rod extends beyond the fulcrum, where the homeowner applies force.
### Formula Application:
To solve for the minimum length \( L \) of the rod, use the principle of torque balance around the fulcrum:
\[
\text{Torque by rock} = \text{Torque by homeowner's force}
\]
\[
m \cdot g \cdot d = F \cdot (L - d)
\]
Where:
- \( m = 325 \, \text{kg} \) is the mass of the rock
- \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity
- \( d = 0.266 \, \text{m} \) is the distance from the fulcrum to the rock
- \( F = 663 \, \text{N} \) is the force applied by the homeowner
### Solution:
Solve the equation for \( L \):
\[
325 \times 9.81 \times 0.266 = 663 \times (L - 0.266)
\]
Determine \( L \) to ensure the lever system efficiently moves the rock.
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