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A block weighs 15 N and is suspended from a spring that is attached to the ceiling. The spring...

Question:

A block weighs 15 N and is suspended from a spring that is attached to the ceiling. The spring stretches by 0.075 m from its unstrained length. By how much does the spring stretch when a 31-N block is suspended from it?

Hooke's Law:

Whenever we attach an object like a metal ball to the bottom end of the vertical spring, we will evidently notice that the ball is lowered by stretching the spring. The amount of stretch produced in spring depends directly on the weight of the hanging metal ball, so the stretched length increases as the weight of the metal ball increases. The metal ball stretches the spring up to a certain extent, and after that, the spring force counterbalances the weight of the ball, and as a result, the static equilibrium condition is achieved. Mathematically:

$$\rm Spring \ force \propto elongation \ of \ the \ spring $$

Answer and Explanation: 1


Given data:

  • {eq}W_1=\rm 15 \ N {/eq} is the weight of the block
  • {eq}x_1=\rm 0.075 \ m {/eq} is the elongation of the spring
  • {eq}k {/eq} is the spring constant
  • {eq}F {/eq} is the spring force


Using Hooke's law, we write:

{eq}F_1=-kx_1 {/eq}


Under static equilibrium:

{eq}\begin{align} \rm \sum F_{vertical}&=0 \\[0.3cm] F_1+W_1&=0 \\[0.3cm] F_1&=-W_1 \end{align} {/eq}


Therefore:

{eq}\begin{align} W_1&=kx_1 \\[0.3cm] k&=\dfrac{W_1}{x_1} \\[0.3cm] &=\rm \dfrac{15 \ N}{0.075 \ m} \\[0.3cm] &=\rm 200 \ N/m \end{align} {/eq}

Therefore, the spring constant of the spring is {eq}\rm 200 \ N/m {/eq}.


The following data was also given:

  • {eq}W_2=\rm 31 \ N {/eq} is the weight of the new block
  • {eq}x_2 {/eq} is the elongation of the spring


Again, using Hooke's law:

{eq}\begin{align} F_2&=kx_2 \\[0.3cm] W_2&=kx_2 \\[0.3cm] x_2&=\dfrac{W_2}{k} \\[0.3cm] &=\rm \dfrac{31 \ N}{200 \ \frac{N}{m}} \\[0.3cm] &\simeq \color{blue}{\boxed { \rm 0.155 \ m}} \end{align} {/eq}

Therefore, the elongation of the spring will be {eq}\color{blue}{\boxed { \rm 0.155 \ m}} {/eq}.



Learn more about this topic:

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Hooke's Law & the Spring Constant: Definition & Equation

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Chapter 4 / Lesson 19
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After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


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