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Chapter 16: Problem 1

Explain how to find the balance point for two people on opposite ends of a (massless) plank that rests on a pivot.

Short Answer

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Question: Two people are standing on a massless plank that rests on a pivot. Person 1 weighs 60 kg, and person 2 weighs 90 kg. The distance (L) between them is 5 meters. Find the balance point distance relative to person 1. Answer: To find the balance point distance relative to person 1, use the formula derived in the solution: x = (W2 * L) / (W1 + W2). Substitute the given values: W1 = 60 kg, W2 = 90 kg, and L = 5 meters. x = (90 kg * 5m) / (60 kg + 90 kg) x = (450 kg * m) / (150 kg) x = 3 meters The balance point distance relative to person 1 is 3 meters.

Step by step solution

01

Identify the given information

There are two people on opposite ends of a massless plank that rests on a pivot. Let us denote the weights of the two people as W1 and W2, and the distances of each person from the pivot as d1 and d2, respectively.
02

Set up the torque equation

Since the plank is in equilibrium (balanced), the net torque about the pivot point is zero. The torque due to each person is given by their weight multiplied by the distance from the pivot point. Therefore, the torque equation can be set up as follows: Torque1 = Torque2, where Torque1 = W1 * d1 and Torque2 = W2 * d2
03

Solve for the balance point distance

From the torque equation, we have: W1 * d1 = W2 * d2 Now, we want to find the balance point for the two people relative to one of the people (let's choose person 1). To do this, let the distance from person 1 to the balance point be x. Then, the distance from person 1 to the pivot point, d1, can be expressed as x, and the distance from person 2 to the pivot point, d2, can be expressed as the distance between person 1 and person 2 on the plank minus x (L - x). Substituting these expressions into the torque equation, we get: W1 * x = W2 * (L - x)
04

Solve for the balance point distance x

To solve for x, we can rearrange the equation and isolate x: W1 * x + W2 * x = W2 * L x (W1 + W2) = W2 * L Now, we can solve for x: x = (W2 * L) / (W1 + W2) The balance point distance, x, is now expressed in terms of the weights of the two people and the distance between them. By substituting the given weight values and the distance between them (L), we can find the balance point distance relative to person 1.

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