Problem 54 - 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. Balancing a Seesaw. Jim and Bob sit at opposite ends of an 18-foot seesaw, with the fulcrum at its center. Jim weighs 160 pounds, and Bob weighs 200 pounds. Kim sits 4 feet in front of Jim, and the seesaw balances. How much does Kim weigh?

Step-by-Step Solution

Answer

Kim weighs 72 pounds.

Step 1: Understand the Lever Balance Equation

The seesaw is in balance, so we apply the principle of levers. The sum of the torques (force multiplied by distance) about the fulcrum on one side must equal the sum on the other side. The general equation for balance is: \[ F_1 \times d_1 + F_2 \times d_2 = F_3 \times d_3 \] where \(F_i\) represents the force (or weight) and \(d_i\) represents the distance from the fulcrum.

Step 2: Assign Known Values

Jim sits at the end of the seesaw, which is 9 feet from the fulcrum (half of 18 feet). Kim is 4 feet in front of Jim, so she is \(9 - 4 = 5\) feet from the fulcrum. Thus, for Jim: - \(F_1 = 160\) pounds - \(d_1 = 9\) feetFor Kim: - \(d_2 = 5\) feet

Step 3: Calculate the Total Torque on Jim's Side

The total torque on Jim and Kim's side is calculated by their weights and distances:\[ 160 \times 9 + K \times 5 \] where \(K\) is Kim's weight, which we need to find.

Step 4: Calculate Torque on Bob's Side

Bob is sitting at the opposite end, 9 feet from the fulcrum, exerting a torque:\[ F_3 = 200 \] pounds\[ d_3 = 9 \] feetThus, the torque he exerts is: \[ 200 \times 9 \]

Step 5: Set Up the Equation and Solve for Kim's Weight

Using the balance condition and the previous calculations:\[ 160 \times 9 + K \times 5 = 200 \times 9 \]Solving for \(K\), we rearrange:\[ 1440 + 5K = 1800 \]Subtract 1440 from both sides:\[ 5K = 360 \]Finally, divide by 5:\[ K = 72 \]

Step 6: Conclusion

Kim weighs 72 pounds. The seesaw is balanced with Jim and Bob sitting at the ends, and Kim sits such that her weight correctly balances the seesaw by the lever principle.

Key Concepts

Torque Calculation

Understanding torque is essential when dealing with levers and seesaws. In physics, torque is considered the rotational equivalent of linear force. It is what causes an object to rotate about a point or axis, which in the case of a seesaw, is the fulcrum. Torque is calculated by multiplying the applied force by the distance from the point of rotation.

In mathematical terms, the torque \( \tau \) can be expressed as:

• \( \tau = F \times d \)

where:

• \( F \) is the force applied, like a person’s weight.
• \( d \) is the distance from the axis or fulcrum.

When balancing a seesaw, the total torque exerted by the people on one side of the seesaw must equal the total torque on the other side. That means, for the seesaw to stay balanced, the weights multiplied by their distances from the fulcrum on one side must match those on the opposing side.

Seesaw Balance

A seesaw, often enjoyed by children in playgrounds, is a practical example of a lever system. Here, both ends of the seesaw experience forces that push downwards due to the weight of the individuals sitting on them. The lever balance equation described in the solution displays the seesaw’s need for equilibrium to achieve balance.

To understand seesaw balance, consider these points:

• The seesaw must have equal torque on both sides to be balanced.
• If one side has more torque, the seesaw will tip, and equilibrium is lost.

In this exercise, Jim and Bob sit on opposite ends of an 18-foot seesaw balanced perfectly due to Kim’s positioning. By calculating the appropriate distances and knowing the weights, Kim's weight was precisely what was needed to maintain balance. This balance showcases how forces and distances can be adjusted to achieve equilibrium.

Fulcrum

The fulcrum acts as the pivotal point in a lever system. In a seesaw, this fulcrum is positioned at the center, supporting the seesaw’s rotation and balance.

In the exercise, the seesaw’s fulcrum is central to maintaining the balance between Jim, Bob, and Kim. The lever is split into equal parts around this fulcrum:

• The distance to each end plays a critical role in determining the torque produced by each person’s weight.
• Jim and Bob are 9 feet from the fulcrum, on either side, balancing each other with the proper counterbalance of Kim’s placement.

Understanding how the fulcrum works helps in visualizing the forces at play. By adjusting distances and force (weight), the lever balance equation comes to life. The fulcrum essentially acts as the balancing point that ensures the seesaw remains stable when the torques align correctly.