Problem 13 - College Physics

Question

Two children are sitting on opposite ends of a uniform seesaw of negligible mass. (a) Can the seesaw be balanced if the masses of the children are different? How? (b) If a \(35-\mathrm{kg}\) child is \(2.0 \mathrm{~m}\) from the pivot point (or fulcrum), how far from the pivot point will her \(30-\mathrm{kg}\) playmate have to sit on the other side for the seesaw to be in equilibrium?

Step-by-Step Solution

Answer

(a) Yes, by adjusting distances. (b) The playmate should sit approximately 2.33 m from the pivot.

Step 1: Determine Balance Condition

For the seesaw to be balanced, the torques on both sides of the pivot must be equal. Torque is calculated as the product of force (due to weight, which is mass times gravity) and distance from the pivot. This can be represented by the equation: \( m_1 \times d_1 = m_2 \times d_2 \).

Step 2: Apply Balance Condition for Different Masses (Part a)

Yes, the seesaw can be balanced even if the children's masses are different. By adjusting their distances from the pivot (so one child is farther from the pivot if they are lighter), the torques can be made equal, thus balancing the seesaw.

Step 3: Calculate Distance for Equilibrium (Part b)

Given: \( m_1 = 35 \, \text{kg}, \) \( d_1 = 2.0 \, \text{m}, \) and \( m_2 = 30 \, \text{kg} \). Use the torque balance equation: \( m_1 \times d_1 = m_2 \times d_2 \). Substitute the known values: \( 35 \, \text{kg} \times 2.0 \, \text{m} = 30 \, \text{kg} \times d_2 \). Solve for \( d_2 \) by dividing both sides by \( 30 \, \text{kg} \), giving \( d_2 = \frac{70}{30} \approx 2.33 \, \text{m} \).

Key Concepts

Torque

Torque is a fundamental concept in physics that refers to the rotational force applied around a pivot point or fulcrum. It can be thought of as a measure of how much a force acting on an object causes that object to rotate. The amount of torque depends on two factors: the magnitude of the force applied and the distance from the pivot point at which it is applied. This distance is often called the lever arm. The formula for torque can be expressed as: \ \( \tau = F \times d \) \ where \( \tau \) is the torque, \( F \) is the force applied, and \( d \) is the distance from the pivot.

• Torque acts in a direction: clockwise or counterclockwise.
• In equilibrium, the sum of clockwise torque must equal the sum of counterclockwise torque.

Understanding this helps us solve problems involving seesaws and other scenarios where rotation about a pivot is significant.

Equilibrium

Equilibrium in physics is the state where all forces and torques are balanced. This means that an object in equilibrium is either at rest or moving at a constant velocity without accelerating. For a seesaw to be in equilibrium, the clockwise torque needs to match exactly the counterclockwise torque. This balance ensures the seesaw remains level and stable, without tipping to one side. There are two types of equilibrium:

• Static Equilibrium: When an object is at rest, and there is no net force or torque acting on it, like a balanced seesaw.
• Dynamic Equilibrium: When an object moves with constant speed, maintaining a consistent state.

In the seesaw exercise, achieving equilibrium required calculations to find the correct distances each child needed to sit from the pivot, ensuring both sides exert equal torques.

Seesaw Balance

A seesaw is a practical example of the torque and equilibrium concepts. Balancing a seesaw involves ensuring that the weight and distance from the pivot on one side equals that on the other side. While it might seem tricky when participants have different weights, balance can still be achieved by compensating with distance. If one child is heavier, they will be closer to the pivot, and the lighter child will sit further away. This configuration ensures that the torques and thus the forces on both sides are similar. By adjusting these parameters, the seesaw can achieve stability and balance.
Using the balance equation \( m_1 \times d_1 = m_2 \times d_2 \), we can calculate where each child needs to sit so that the seesaw does not tip over.

Lever Principle

The lever principle is key to understanding balance scenarios, like on a seesaw. This principle states that a lever amplifies an input force to provide a greater output force, allowing heavier objects or a load to be lifted or balanced. The seesaw acts as a lever, with each child representing an input force and the pivot acting as the fulcrum. The lever principle can be summarized as follows:

• Effort force multiplied by its distance from the fulcrum is equal to the load force multiplied by its distance from the fulcrum.
• This means balance can still be struck even if the weights (forces) differ, by varying their distances from the fulcrum.

This concept is a building block in physics, illustrating how small force changes at greater distances can create equilibrium. Using calculations, a balance on the seesaw can be achieved by applying the lever principle effectively.