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Chapter 12: Problem 18

(II) 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 \(3.2 \mathrm{~m}\) long (Fig. \(12-57\) ). Two playmates are already on either end. Boy \(A\) has a mass of \(45 \mathrm{~kg}\), and boy \(\mathrm{B}\) a mass of \(35 \mathrm{~kg} .\) Where should girl \(\mathrm{C},\) whose mass is \(25 \mathrm{~kg}\), place herself so as to balance the seesaw?

Short Answer

Expert verified
Girl C should sit 0.64 meters from the center towards Boy B to balance the seesaw.

Step by step solution

01

Understand the problem

The seesaw is a beam that is 3.2 meters long with a fulcrum in the center. Boy A is at one end, and Boy B is at the other. We need to find the position of Girl C, who weighs 25 kg, to balance the seesaw.
02

Identify key points and distances

The key points are Boy A's position at one end, at a distance of 1.6 meters from the fulcrum (midpoint of the seesaw), and Boy B's position, also 1.6 meters from the fulcrum, at the opposite end. Girl C needs to be placed somewhere between or beyond these two points.
03

Apply the principle of moments

Balance is achieved when the clockwise moments equal the counterclockwise moments. The moment is calculated as mass times distance from the pivot point: \[ \text{Moment} = \text{mass} \times \text{distance} \]
04

Setup the equation for balance

The moments of Boys A and B (Boy A): \[ 45 \times 1.6 \] (Boy B): \[ 35 \times 1.6 \] These moments provide: \[ 45 \cdot 1.6 = 72 \quad \text{and} \quad 35 \cdot 1.6 = 56 \]
05

Find the position of Girl C using moments

To balance the seesaw, the sum of the clockwise moments must equal the sum of the counterclockwise moments. Let Girl C's position relative to the fulcrum be \(d\): \[ 25 \cdot d = 72 - 56 \] Simplifying gives: \[ 25d = 16 \]
06

Solve the equation for Girl C's position

Solving for \(d\): \[ d = \frac{16}{25} = 0.64 \] Girl C should sit 0.64 meters to the left of the center (toward Boy B) to balance the seesaw.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Force
When we speak about the moment of force, we're talking about the turning effect that a force has on an object. Imagine trying to open a door; you apply force on the handle to create a turning effect around the hinges. This is the moment of force in action.
The moment of force is calculated by multiplying the force applied (which could be a person's weight) by the distance from the pivot point or fulcrum. In mathematical terms, this is expressed as:
  • Moment = mass 脳 distance
In our seesaw problem, each child's weight, multiplied by their distance from the seesaw's fulcrum, gives us the moment of force they exert on the seesaw. The aim is to balance these moments on either side of the fulcrum to achieve equilibrium.
Seesaw Equilibrium
Equilibrium on a seesaw is a magical balance state where the seesaw is perfectly level. To reach this state, the seesaw must have equal forces acting on both sides of the fulcrum. Essentially, the clockwise moments have to be equal to the counterclockwise moments.
This means if one side has a heavier weight sitting farther from the fulcrum, the other side needs either a heavier weight or the same weight placed farther from the fulcrum to balance out the moments:
  • Clockwise moments = Counterclockwise moments
In our scenario, Boy A produces a clockwise moment due to his position, while Boy B's position produces a counterclockwise moment. We find balance by determining where Girl C should sit, so her weight contributes to the counterclockwise moment, bringing the system into equilibrium.
Fulcrum Pivot Point
The fulcrum is the heart of any simple lever like a seesaw. It's the central pivot point where the seesaw board balances and rotates. Picture it as the middle point under the seesaw where the board tilts but should ideally not tip over.
The distance of each child from the fulcrum directly influences the moment of force they can exert on the board. By changing their position relative to the fulcrum, they can either increase or decrease their moment of force. Thus, the fulcrum is pivotal (pun intended!) to achieving balance.
  • It's the point where all moments around it should add up to zero, ensuring stable balance in moments.
In our exercise, identifying the correct point from the fulcrum where Girl C should position herself is key to balancing the moments on the seesaw.

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Most popular questions from this chapter

(II) A 110 -kg horizontal beam is supported at each end. A 320 -kg piano rests a quarter of the way from one end. What is the vertical force on each of the supports?

(III) A cubic crate of side \(s=2.0 \mathrm{m}\) is top-heavy: its \(\infty \mathrm{is} 18 \mathrm{cm}\) above its true center. How steep an incline can the crate rest on without tipping over? What would your answer be if the crate were to slide at constant speed down the plane without tipping over? [Hint. The normal force would act at the lowest corner.]

A tightly stretched "high wire" is \(36 \mathrm{~m}\) long. It sags \(2.1 \mathrm{~m}\) when a 60.0 -kg tightrope walker stands at its center. What is the tension in the wire? Is it possible to increase the tension in the wire so that there is no sag?

(II) If a compressive force of \(3.3 \times 10^{4} \mathrm{~N}\) is exerted on the end of a \(22-\mathrm{cm}\) -long bone of cross-sectional area \(3.6 \mathrm{~cm}^{2},(a)\) will the bone break, and \((b)\) if not, by how much does it shorten?

A \(23.0-\mathrm{kg}\) backpack is suspended midway between two trees by a light cord as in Fig. \(12-50 .\) A bear grabs the backpack and pulls vertically downward with a constant force, so that each section of cord makes an angle of \(27^{\circ}\) below the horizontal. Initially, without the bear pulling, the angle was \(15^{\circ}\); the tension in the cord with the bear pulling is double what it was when he was not. Calculate the force the bear is exerting on the backpack.
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