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 95e

Chapter 12, Problem 95e

Suppose a 65-kg person jumps from a height of 3.0 m down to the ground. Estimate the stress and determine if the tibia will break in a stiff-legged landing (d = 1.0 cm).

Verified step by step guidance

Determine the force exerted on the tibia during the landing. Start by calculating the velocity just before impact using the equation for free fall:

\sqrt{2 g h}

, where g is the acceleration due to gravity (9.8 m/s²) and h is the height (3.0 m).

Estimate the deceleration during the landing. Assume the stopping distance (compression of the tibia) is approximately equal to the diameter of the tibia,

d = 1.0 cm = 0.01 m

. Use the kinematic equation

v^{2} = 2 a d

to solve for the deceleration

a

Calculate the force exerted on the tibia using Newton's second law:

F = m a

, where

m

is the mass of the person (65 kg) and

a

is the deceleration calculated in the previous step.

Determine the stress on the tibia. Stress is defined as force per unit area:

σ = F / A

. The cross-sectional area of the tibia can be approximated as a circle with diameter

d = 1.0 cm

, so

A = π(d /2)

².

Compare the calculated stress to the breaking stress of the tibia (approximately 1.6 × 10⁸ N/m²). If the calculated stress exceeds this value, the tibia will break; otherwise, it will not.

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Key Concepts

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

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For a person jumping from a height, this energy can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. As the person falls, this potential energy is converted into kinetic energy, which is crucial for understanding the forces involved upon landing.

Impact Force

Impact force is the force exerted by an object when it comes to a sudden stop after falling. It can be estimated using the impulse-momentum theorem, which relates the change in momentum to the force applied over the time of impact. In a stiff-legged landing, the impact force is concentrated over a short distance, leading to higher stress on the bones, which is essential for assessing the risk of injury.

Stress and Strain

Stress is defined as the force applied per unit area within materials, while strain is the deformation that occurs as a result of that stress. In the context of the tibia during a landing, calculating the stress involves determining the impact force and the cross-sectional area of the bone. Understanding these concepts helps in evaluating whether the stress exceeds the bone's strength, which could lead to a fracture.

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Related Practice

Textbook Question

A uniform beam of mass M and length ℓ is mounted on a hinge at a wall as shown in Fig. 12–101. It is held in a horizontal position by a wire making an angle θ as shown. A mass m is placed on the beam a distance x from the wall, and this distance can be varied. Determine, as a function of x, the components of the force exerted by the beam on the hinge.

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Textbook Question

A steel rod of radius R = 15 cm and length ℓ₀ stands upright on a firm surface. A 78-kg man climbs atop the rod. When a metal is compressed, each atom throughout its bulk moves closer to its neighboring atom by exactly the same fractional amount. If iron atoms in steel are normally 2.0 x 10⁻¹⁰ m apart, by what distance did this interatomic spacing have to change in order to produce the normal force required to support the man? [Note: Neighboring atoms repel each other, and this repulsion accounts for the observed normal force.]

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Textbook Question

If 25 kg is the maximum mass m that a person can hold in a hand when the arm is positioned with a 105° angle at the elbow as shown in Fig. 12–102, what is the maximum force Fₘₐₓ that the biceps muscle exerts on the forearm? Assume the forearm and hand have a total mass of 2.0 kg with a cg that is 15 cm from the elbow, and that the biceps muscle attaches 5.0 cm from the elbow.

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