Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 14: Periodic Motion

문제 36

14장, 문제 36

A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

검증된 단계별 안내

Start by understanding the energy conservation in a spring-mass system. The total mechanical energy in the system is the sum of kinetic energy (KE) and elastic potential energy (PE). At any point in the oscillation, this total energy remains constant.

Recall the formulas for kinetic energy and elastic potential energy in a spring system. The kinetic energy is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. The elastic potential energy is given by \( PE = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

Since the problem states that the elastic potential energy equals the kinetic energy, set \( \frac{1}{2}mv^2 = \frac{1}{2}kx^2 \). This equation implies that the energy is equally distributed between kinetic and potential forms at this point in the oscillation.

Simplify the equation \( mv^2 = kx^2 \) to find the relationship between \( x \) and \( A \). Recall that the maximum displacement \( A \) is the amplitude of the oscillation, and at maximum displacement, all energy is potential, \( PE = \frac{1}{2}kA^2 \).

Solve for \( x \) in terms of \( A \) by recognizing that when \( PE = KE \), the displacement \( x \) is such that \( x^2 = \frac{A^2}{2} \). Therefore, \( x = \frac{A}{\sqrt{2}} \). This is the distance from the equilibrium position where the energies are equal.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simple Harmonic Motion

Simple harmonic motion describes the oscillatory motion of a mass attached to a spring, where the restoring force is proportional to the displacement from the equilibrium position. The motion is sinusoidal, characterized by amplitude, frequency, and phase, and is governed by Hooke's Law.

Elastic Potential Energy

Elastic potential energy in a spring system is the energy stored due to its deformation, calculated as (1/2)kx^2, where k is the spring constant and x is the displacement from equilibrium. This energy is maximum at the amplitude and zero at the equilibrium position.

Kinetic Energy in Oscillations

Kinetic energy in oscillatory motion is the energy due to the mass's velocity, given by (1/2)mv^2. In a spring-mass system, kinetic energy is maximum at the equilibrium position and zero at the amplitude, varying inversely with elastic potential energy during oscillation.

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