Ch. 12 - Static Equilibrium; Elasticity and Fracture

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 12 - Static Equilibrium; Elasticity and Fracture

Problem 20

Chapter 12, Problem 20

A 172-cm-tall person lies on a light (massless) board which is supported by two scales, one under the top of her head and one beneath the bottom of her feet (Fig. 12–65). The two scales read, respectively, 35.1 and 31.6 kg. What distance is the center of gravity of this person from the top of her head?
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Step 1: Understand the problem. The person is lying on a massless board supported by two scales. The scales measure the forces exerted at the head and feet. The goal is to find the distance of the person's center of gravity from the top of their head. This is a torque equilibrium problem.

Step 2: Define the variables. Let the total weight of the person be \( W = (35.1 + 31.6) \, \text{kg} \times 9.8 \, \text{m/s}^2 \). Let \( x \) be the distance of the center of gravity from the top of the head. The total length of the person is \( L = 1.72 \, \text{m} \).

Step 3: Apply the principle of torque equilibrium. For the board to be in equilibrium, the sum of torques about any point must be zero. Choose the bottom of the feet as the pivot point. The torque due to the weight at the center of gravity is \( W \cdot x \), and the torque due to the scale at the head is \( 35.1 \cdot 9.8 \cdot L \).

Step 4: Write the torque equilibrium equation. The equation is \( 35.1 \cdot 9.8 \cdot L = W \cdot x \). Substitute \( W = (35.1 + 31.6) \cdot 9.8 \) and solve for \( x \).

Step 5: Simplify the equation to find \( x \). Rearrange the equation to isolate \( x \): \( x = \frac{35.1 \cdot 9.8 \cdot L}{(35.1 + 31.6) \cdot 9.8} \). Cancel out \( 9.8 \) and substitute \( L = 1.72 \, \text{m} \) to find the distance of the center of gravity from the top of the head.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Center of Gravity

The center of gravity is the point at which the total weight of an object is considered to act. For a uniform object, it is located at its geometric center, but for irregular shapes or distributions of mass, it can be calculated based on the distribution of weight. In this problem, understanding the center of gravity is crucial for determining how the weight is distributed along the person's height.

Static Equilibrium

Static equilibrium occurs when an object is at rest and the sum of all forces and torques acting on it is zero. In this scenario, the person lying on the board is in static equilibrium, meaning the forces exerted by the scales must balance the gravitational force acting on the person. This concept is essential for analyzing the forces involved and calculating the position of the center of gravity.

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point. In this case, the scales act as pivot points, and the torques created by the weights at the head and feet must balance for the person to remain in equilibrium. Understanding torque is necessary to find the distance from the head to the center of gravity.

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