Three children are trying to balance on a seesaw, which includes a fulcrum rock acting as a pivot at the center, and a very light board 3.2 m long (Fig. 12–60). Two playmates are already on either end. Boy A has a mass of 45 kg, and boy B a mass of 35 kg. Where should girl C, whose mass is 25 kg, place herself so as to balance the seesaw?

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Understand the problem: The seesaw is in rotational equilibrium, meaning the sum of the torques about the fulcrum must be zero. Torque is calculated as the product of force (weight) and the perpendicular distance from the pivot point. The goal is to find the position of girl C such that the torques on both sides of the fulcrum balance.

Define the system: The seesaw is 3.2 m long, so the fulcrum is at the center, dividing the board into two halves of 1.6 m each. Boy A is at one end (1.6 m from the fulcrum), and boy B is at the other end (also 1.6 m from the fulcrum). Girl C's position is unknown, and we will denote her distance from the fulcrum as \( x \).

Write the torque equation: The clockwise torque caused by boy A must equal the counterclockwise torque caused by boy B and girl C. Using \( \tau = F \cdot d \), where \( F \) is the weight (mass \( \cdot \) gravity) and \( d \) is the distance from the fulcrum, the equation becomes: \( (45 \cdot 9.8 \cdot 1.6) = (35 \cdot 9.8 \cdot 1.6) + (25 \cdot 9.8 \cdot x) \).

Simplify the equation: Cancel out the common factor of \( 9.8 \) (gravitational acceleration) from all terms, leaving: \( 45 \cdot 1.6 = 35 \cdot 1.6 + 25 \cdot x \). Expand and simplify further to isolate \( x \): \( 72 = 56 + 25x \).

Solve for \( x \): Subtract \( 56 \) from both sides to get \( 16 = 25x \). Then divide both sides by \( 25 \) to find \( x \): \( x = \frac{16}{25} \). This gives the distance girl C should place herself from the fulcrum to balance the seesaw.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Torque

Torque is a measure of the rotational force applied around a pivot point, calculated as the product of the force and the distance from the pivot. In the context of a seesaw, each child's weight creates a torque about the fulcrum, which must be balanced for the seesaw to remain level. The formula for torque is τ = r × F, where τ is torque, r is the distance from the pivot, and F is the force (weight in this case).

Equilibrium

Equilibrium in physics refers to a state where all forces and torques acting on an object are balanced, resulting in no net force or rotation. For the seesaw to be in equilibrium, the sum of the torques produced by the children on either side of the fulcrum must be equal. This principle allows us to determine where girl C should sit to achieve balance.

Center of Mass

The center of mass is the point at which the mass of an object is evenly distributed in all directions. In a seesaw scenario, the position of each child relative to the fulcrum affects the overall center of mass of the system. Understanding how to locate the center of mass helps in determining the correct position for girl C to balance the seesaw with the other two children.