Stretching a Spring A spring has a natural length of 10 in....

Name: Problem 8, Chapter 7: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2019-07-18T15:20:57-08:00
Duration: 11 min 12 s
Channel: Carlos Pinilla
Description: Stretching a Spring A spring has a natural length of 10 in. An 800 -lb force stretches the spring to 14 in. (a) Find the force constant. (b) How much work is done in stretching the spring from 10 in. to 12 in.? (c) How far beyond its natural length will a 1600 -lb force stretch the spring?

Stretching a Spring A spring has a natural length of 10 in. An 800 -lb force stretches the spring to 14 in. (a) Find the force constant. (b) How much work is done in stretching the spring from 10 in. to 12 in.? (c) How far beyond its natural length will a 1600 -lb force stretch the spring?

Stretching a Spring A spring has a natural length of 10 in. An 800 -lb force stretches the spring to 14 in.
(a) Find the force constant.
(b) How much work is done in stretching the spring from 10 in. to 12 in.?
(c) How far beyond its natural length will a 1600 -lb force stretch the spring?

Key Concepts

Hooke's Law

Hooke's Law is the principle that states that the force needed to extend or compress a spring by some distance is proportional to that distance. It provides the foundational relationship of F = kx, where F is the applied force, x is the displacement from the spring's natural length, and k is the spring constant. This law is essential for analyzing forces on elastic materials.

Spring Constant

The spring constant, denoted by k, measures the stiffness of a spring. It is calculated by dividing the force applied by the displacement produced, following the relationship k = F/x. Determining the spring constant is crucial for predicting how a spring will behave under various loads.

Work Done in Stretching a Spring

The work done in stretching (or compressing) a spring involves calculating the energy required to deform it. This work is obtained by integrating the variable force over the displacement, resulting in the formula W = (1/2)k(x_final^2 - x_initial^2). This concept connects energy with the mechanical properties of a spring.

Integration in Variable Force Problems

Integration is used to calculate work when the force is not constant, such as in the case of a spring where the force changes linearly with displacement. By integrating the force function (derived from Hooke's Law) over the range of displacement, one can determine the total work done in stretching or compressing the spring.

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