Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 17

Chapter 10, Problem 17

A stretched spring stores 2.0 J of energy. How much energy will be stored if the spring is stretched three times as far?

Verified step by step guidance

Understand that the energy stored in a spring is given by the formula:

E_{s} = \frac{1}{2} k x^{2}

, where

k

is the spring constant and

x

is the displacement from the equilibrium position.

Recognize that the energy stored in the spring is proportional to the square of the displacement, as indicated by the

x^{2}

term in the formula.

If the spring is stretched three times as far, the new displacement becomes

x' = 3 x

. Substitute this into the energy formula:

E_{s}' = \frac{1}{2} k {(3x)}^{2}

Simplify the expression for the new energy:

E_{s}' = \frac{1}{2} k (9 x^{2}) = 9 \frac{1}{2} k x^{2}

Compare the new energy to the original energy. Since the original energy is

E_{s} = \frac{1}{2} k x^{2}

, the new energy is 9 times the original energy. Multiply the original energy (2.0 J) by 9 to find the new energy.

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Key Concepts

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

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed from its equilibrium position, represented mathematically as F = kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave under different loads.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is stretched or compressed. The formula for elastic potential energy (U) is U = 1/2 kx², where k is the spring constant and x is the displacement from the equilibrium position. This concept is crucial for calculating the energy stored in a spring based on its deformation.

Energy Scaling with Displacement

The energy stored in a spring scales with the square of the displacement. This means that if the displacement is tripled, the energy stored increases by a factor of nine (since (3x)² = 9x²). Understanding this relationship is essential for predicting how much energy will be stored when the spring is stretched to different lengths.

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Related Practice

Textbook Question

The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

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Textbook Question

As a 15,000 kg jet plane lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 60,000 N/m. If the spring stretches 30 m to stop the plane, what was the plane's landing speed?

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Textbook Question

The maximum energy a bone can absorb without breaking is surprisingly small. Experimental data show that a leg bone of a healthy, 60 kg human can absorb about 200 J. From what maximum height could a 60 kg person jump and land rigidly upright on both feet without breaking his legs? Assume that all energy is absorbed by the leg bones in a rigid landing.

Textbook Question

FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

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Textbook Question

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. Suppose the dam is 80% efficient at converting the water's potential energy to electrical energy. How many kilograms of water must pass through the turbines each second to generate 50 MW of electricity? This is a typical value for a small hydroelectric dam.

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Textbook Question

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is $Δ U_{G}$ of 1.0 kg of water?