Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 14: Periodic Motion

Problem 34

Chapter 14, Problem 34

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?

Verified step by step guidance

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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Key Concepts

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

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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Textbook Question

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Textbook Question

A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute the total mechanical energy of the glider at any point in its motion

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A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is 0.250 m and the period is 3.20 s. What are the speed and acceleration of the block when x = 0.160 m?

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