Problem 51 - Intermediate Algebra

Question

Use the following property of levers: \(A\) lever will be in balance when the sum of the products of the forces on one side of a fulcrum and their respective distances from the fulcrum is equal to the sum of the products of the forces on the other side of the fulcrum and their respective distances from the fulcrum. Moving a Stone. A woman uses a 10 -foot bar to lift a 210 -pound stone. If she places another rock 3 feet from the stone to act as the fulcrum, how much force must she exert to move the stone?

Step-by-Step Solution

Answer

The woman needs to exert 90 pounds of force.

Step 1: Understanding the Lever Principle

A lever is in balance when the sum of the products of the force applied and its distance from the fulcrum on one side equals the sum of the products on the other side. This is represented by the equation \( F_1 imes d_1 = F_2 imes d_2 \), where \( F \) represents force and \( d \) represents distance from the fulcrum.

Step 2: Identifying the Forces and Distances

We are given a woman lifting a 210-pound stone using a lever. The stone is 3 feet from the fulcrum. Therefore, \( F_1 = 210 \) pounds and \( d_1 = 3 \) feet. The total length of the bar is 10 feet, so \( d_2 = 10 - 3 = 7 \) feet, representing the distance from the fulcrum to where the woman applies her force \( F_2 \).

Step 3: Setting up the Equation

Using the balance equation \( F_1 \times d_1 = F_2 \times d_2 \), substitute the known values to form the equation: \( 210 \times 3 = F_2 \times 7 \).

Step 4: Solving for the Unknown Force

Calculate the left-hand side: \( 210 \times 3 = 630 \). Set this equal to \( F_2 \times 7 \), giving us the equation: \( 630 = F_2 \times 7 \). Divide both sides by 7 to solve for \( F_2 \): \( F_2 = \frac{630}{7} = 90 \).

Step 5: Conclusion with Required Force

The woman needs to exert a force of 90 pounds to lift the stone using the lever.

Key Concepts

Mechanical Advantage

The term "mechanical advantage" refers to the benefit gained by using a tool or machine to amplify the input force. In the context of levers, mechanical advantage allows a person to lift heavy objects with less effort than it would otherwise take.
Levers are simple machines designed to provide mechanical advantage. By positioning the fulcrum at strategic points along the lever, the effort required to lift a load can be significantly reduced.

• When the fulcrum is closer to the load than the point where the force is applied, the mechanical advantage is increased.
• This means the effort needed is less than the weight of the load.

Thus, in our example, the woman used a long bar to magnify her force of 90 pounds to lift a 210-pound stone. This illustrates how cleverly arranged levers can help us perform tasks that would otherwise require much greater force.

Forces and Distances in Levers

In the study of levers, two main components must be considered: forces and distances.
The force exerted depends on how far from the fulcrum this force is applied. These two factors are intertwined: as the distance from the fulcrum increases, the amount of force required decreases, assuming the lever is balanced.

• We define these forces using the equation: \( F_1 \times d_1 = F_2 \times d_2 \).
• Where \( F_1 \) and \( F_2 \) are the forces applied, and \( d_1 \) and \( d_2 \) are their respective distances from the fulcrum.

For example, in the exercise, the woman applies a force 7 feet from the fulcrum, in contrast to the stone's position of 3 feet from the fulcrum. Such an arrangement allows her to exert considerably less force to lift the stone.

Fulcrum in Physics

The fulcrum is the pivotal point around which a lever rotates. It's the anchor holding the lever in place, providing the balance point between effort and load.
The placement of the fulcrum is crucial as it determines the lever's effectiveness.

• Positioning the fulcrum closer to the load reduces the force needed to lift it.
• Conversely, placing the fulcrum farther from the load increases the effort needed, reducing mechanical advantage.

In our exercise, the fulcrum was placed 3 feet from the stone. This setup allowed the woman to exert a force on the opposite end that was fewer pounds than the stone, effectively balancing the lever according to the lever balance principle.

Newton's Laws of Motion

Newton's laws of motion are foundational to understanding how levers and other mechanical systems work. In the context of our exercise, they help explain how and why forces are balanced.
Newton's first law, the law of inertia, states that an object will stay at rest or in motion unless acted upon by a force. Applying a force on one side of the lever causes the stone on the other side to move.
Newton's second law relates force to mass and acceleration, given by the formula \( F = ma \). This law helps us understand the need for a certain amount of force to move the stone.

• It's this principle that underscores why the woman can move the stone with a smaller force due to the mechanical advantage provided by the lever.

Revisiting these laws in practice means recognizing how forces applied at various points on a lever result in motion, a true embodiment of physics in everyday life.