Question

When a $0.213-\mathrm{kg}$ mass is attached to a vertical spring, it causes the spring to stretch a distance $d$. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, $d$.

Name: Problem 43, Chapter 13: Physics
Uploaded: 2021-03-24T05:59:05-08:00
Duration: 1 min 28 s
Channel: Narayan Hari
Description: When a 0.213-kg mass is attached to a vertical spring, it causes the spring to stretch a distance d. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, d.

   When a $0.213-\mathrm{kg}$ mass is attached to a vertical spring, it causes the spring to stretch a distance $d$. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, $d$.

Physics

James S. Walker 4th Edition

Chapter 13, Problem 43

Instant Answer

Solved by Expert Narayan Hari

03/24/2021

Step 1

Step 1: We know that the time period of oscillation is given by $T = 2\pi\sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant. Show more…

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When a $0.213-\mathrm{kg}$ mass is attached to a vertical spring, it causes the spring to stretch a distance $d$. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, $d$.

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Key Concepts

Hooke's Law

This principle relates the force exerted by a spring to its displacement from equilibrium, stating that the force is proportional to the displacement (F = kx). The constant of proportionality, k, is known as the spring constant, and it characterizes the stiffness of the spring.

Equilibrium in a Mass-Spring System

When a mass is attached to a vertical spring, the downward gravitational force on the mass (mg) is balanced by the upward restoring force of the spring (k·d) at equilibrium. This balance allows for the determination of the static stretch distance of the spring as d = mg/k.

Simple Harmonic Motion (SHM)

In a mass-spring system, when the mass is slightly displaced from its equilibrium position, it undergoes simple harmonic motion. The period of oscillation is given by T = 2??(m/k), which connects the mass and the spring constant, and is used to analyze the timing of the oscillatory motion, linking dynamic behavior with the system's properties.

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