Question

A \(5.00-\mathrm{m}\) -long board of mass \(50.0 \mathrm{~kg}\) is used as a seesaw. On the left end of the seesaw sits a 45.0 -kg girl, and on the right end sits a 60.0 -kg boy. Determine the position of the pivot point for static equilibrium.

Short answer

Answer: The pivot point should be placed approximately 1.67 meters from the left end of the seesaw for static equilibrium.

Step-by-step solution

Define variables and write the formula for static equilibrium

Let's define the variables: The weight of the board (W_board) = 50 kg The weight of the girl (W_girl) = 45 kg The weight of the boy (W_boy) = 60 kg The length of the seesaw (L) = 5 m Distance of the pivot point to the left end (x_pivot) In static equilibrium, the total clockwise torque equals the total counterclockwise torque. Thus, the formula for the equilibrium is: torque_girl + torque_board = torque_boy

Calculate the torques

To calculate the individual torques, we will multiply the weights by their respective distances from the pivot point. For the girl, the distance is equal to the distance of the pivot point to the left end (x_pivot). For the board, we account for the uniform distribution of weight across the seesaw and use half of the length of the seesaw as the distance from the pivot (0.5 * L). For the boy, the distance is the total length of the seesaw minus the distance of the pivot from the left end (L - x_pivot). torque_girl = W_girl * x_pivot torque_board = W_board * (0.5 * L) torque_boy = W_boy * (L - x_pivot)

Solve for x_pivot

Now, plug the torques into the static equilibrium formula and solve for x_pivot: W_girl * x_pivot + W_board * (0.5 * L) = W_boy * (L - x_pivot) 45 * x_pivot + 50 * (0.5 * 5) = 60 * (5 - x_pivot)

Simplify the equation

Simplify the equation: 45 * x_pivot + 125 = 300 - 60 * x_pivot Now, combine the x_pivot terms: 105 * x_pivot = 175 Now, divide by 105 to find the value of x_pivot: x_pivot = 175 / 105 x_pivot ≈ 1.67 m The position of the pivot point for static equilibrium is approximately 1.67 meters from the left end of the seesaw.

Key concepts

Torque

Torque is a fundamental concept in physics that describes the rotational effect of a force applied at a distance from a pivot point. The magnitude of torque (\( \tau \)) is calculated by the product of the force (F) applied and the distance (d) from the pivot point:\[\tau = F \times d\]
In the context of a seesaw, torque determines how rotational forces balance to keep the system in static equilibrium. If torques on either side of the pivot are equal, the seesaw doesn't rotate, and it stays in balance. This equality is crucial for solving problems involving seesaws or any lever system in physics.

• Clockwise Torque: Torque causing a rotation in the clockwise direction.
• Counterclockwise Torque: Torque causing rotation in the counterclockwise direction.

Understanding the direction and magnitude of each torque can help solve static equilibrium problems like determining the correct pivot point on a seesaw.

Seesaw Problems

Seesaw problems are a classic example in physics for illustrating principles of torque and static equilibrium. In these problems, you are typically asked to ensure that the seesaw is perfectly balanced, which means that the total torque about the pivot point must equal zero.
To tackle seesaw problems, you need to consider:

• Weights on each side: Different masses can be placed at different distances, creating varying torques.
• Pivot point: This is the fulcrum or central point where the seesaw balances. Finding the correct position is key to maintaining balance.

The challenge is ensuring the torques generated by weights on either side of the pivot are equal. Solving these problems can aid in understanding real-world applications, like calculating where support beams should be placed under a bridge or playground equipment.

Physics Problems

In physics, problems like the seesaw exercise are used to apply theoretical principles to practical situations. These problems often blend multiple physics concepts including forces, torque, and equilibrium to demonstrate how systems function under real-world conditions.
Physics problems encourage:

• Critical Thinking: Analyzing different forces and distances to find solutions.
• Problem-Solving Skills: Applying formulas and principles like those of torque and balance.

The ultimate goal of problems like these is to convey an understanding of how various forces interact and balance in everyday systems like seesaws. This helps in predicting outcomes and designing systems that can withstand everyday use without failing.

Lever Principle

The lever principle is fundamental in understanding balance and torque. A lever works by amplifying an input force to lift or move heavy loads, and its effectiveness is governed by the positioning of the effort force, load, and fulcrum.
The key aspects of the lever principle include:

• Effort Arm: The distance from the fulcrum to the effort force, crucial for creating torque.
• Load Arm: The distance from the fulcrum to the load, influencing how the load is managed by the lever.
• Fulcrum: The pivot point that determines the leverage and balance of forces.

According to the lever principle, for balance in a system, the torques exerted by the effort and the load around the fulcrum must be equal. In a seesaw scenario, even if the distances or weights differ between sides, balance can still be achieved through strategic placement of the pivot. This understanding helps in designing mechanisms that require precise weight balancing and distribution.