Ch 15: Oscillations

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 15: Oscillations

Problem 41a

Chapter 15, Problem 41a

When the displacement of a mass on a spring is (½)A, what fraction of the energy is kinetic energy and what fraction is potential energy?

Verified step by step guidance

Start by recalling the total mechanical energy of a mass-spring system, which is the sum of kinetic energy (KE) and potential energy (PE). The total energy is given by:

E = U^{2} + K^{2}

, where

U

is potential energy and

K

kinetic energy

Recall the potential energy in a spring is given by:

U = 1/2 k^{2}

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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 its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant. This principle is fundamental in understanding how springs store potential energy when displaced.

Potential Energy in Springs

The potential energy (PE) stored in a spring is given by the formula PE = 1/2 kx², where x is the displacement from the equilibrium position. This energy is maximum when the spring is fully compressed or stretched and decreases as the spring returns to its equilibrium position.

Kinetic Energy

Kinetic energy (KE) is the energy of an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In the context of a mass on a spring, kinetic energy is highest when the mass passes through the equilibrium position, where potential energy is zero.

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Textbook Question

A 350 g mass on a 45-cm-long string is released at an angle of 4.5° from vertical. It has a damping constant of 0.010 kg/s. After 25 s, (a) how many oscillations has it completed and (b) what fraction of the initial energy has been lost?

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