A particular spring has a force constant of 2.5 x 10^3 N/m. (A) How much work is done in...

Question:

A particular spring has a force constant of 2.5 x 10{eq}^3 {/eq} N/m.

(A) How much work is done in stretching the relaxed spring by 6.0 cm?

(B) How much more work is done in stretching the spring an additional 2.0 cm?

Hooke's Law:

In Physics, Hooke's law defines the relationship between the force required to compress the spring and the stiffness of the spring. When a spring stretched from its initial position {eq}(a) {/eq} to final position {eq}(b) {/eq}, the work done by the spring can be calculated by using the following equation:

$$W=\int_{a}^{b} F{\mathrm{d} x}$$

• F is the force required to compress the spring {eq}(\text{N}) {/eq}

We are given the following data:

• Force constan {eq}k=2.5\times10^{3}\ \text{N/m} {/eq}

A) Work done in stretching the relaxed spring by 6.0 cm is

We can calculate the work done in stretching the spring form a relaxed position {eq}(x=0) {/eq} to {eq}(x=6.0\ \text{cm}) {/eq} by using the following relation:

\begin{align} W&=\int_{a}^{b} F{\mathrm{d} x}\\[0.3 cm] &=\int_{0}^{0.06}\left (kx \right ){\mathrm{d} x}&\left [ F=kx \right ]\\[0.3 cm] &=\left [ \dfrac{kx^{2}}{2} \right ]_{0}^{0.06}&\left [\text{1 cm=0.01 m} \right ]\\[0.3 cm] &=2.5\times10^{3}times\left ( 0.06^{2}-0^{2} \right )\\[0.3 cm] &=9.0\ \text{N.m}&\left [ \text{1 N.m=1 J} \right ]\\[0.3 cm] &=\boxed{\color{blue}{9.0\ \text{J}}} \end{align}

(B) Work done in stretching the spring by an additional 2.0 cm is:

Spring is stretched from {eq}(x'=6.0\ \text{cm}) {/eq} to {eq}(x'=6.0+2.0=8.0\ \text{cm}) {/eq}

\begin{align} W'&=\int_{a}^{b} F' {\mathrm{d} x}\\[0.3 cm] &=\int_{0.06}^{0.08}\left (kx' \right ){\mathrm{d} x}&\left [ F'=kx' \right ]\\[0.3 cm] &=\left [ \dfrac{k' x' ^{2}}{2} \right ]_{0.06}^{0.08}&\left [\text{1 cm=0.01 m} \right ]\\[0.3 cm] &=2.5\times10^{3}\times\left ( 0.08^{2}-0.06^{2} \right )\\[0.3 cm] &=7\ \text{N.m}&\left [ \text{1 N.m=1 J} \right ]\\[0.3 cm] &=\boxed{\color{blue}{7\ \text{J}}} \end{align}

Practice Applying Spring Constant Formulas

from

Chapter 17 / Lesson 11
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In this lesson, you'll have the chance to practice using the spring constant formula. The lesson includes four problems of medium difficulty involving a variety of real-life applications.