Suppose that 4 J of work is needed to stretch a spring from its natural length of 36 cm to a...

Question:

Suppose that 4 J of work is needed to stretch a spring from its natural length of 36 cm to a length of 45 cm.

(a) How much work is needed to stretch the spring from 41 cm to 43 cm? _____J

(b) How far beyond its natural length will a force of 40 N keep the spring stretched?_____cm

Energy Stored in Spring

When a force is applied to a spring, it gets stretched from its original length and stores some energy into it. The amount of energy stored in the spring during its stretching is called the strain energy.

Answer and Explanation: 1


Let k is the stiffness of the spring

As per the statement,4 J of work is needed to stretch a spring from 36 cm to 45 cm

The change in length of the spring is {eq}\Delta \text{x}=45-36=9\ \text{cm} {/eq}

By using Hooke's Law

$$\begin{align} W&=\dfrac{1}{2}\times k \times (\Delta x)^{2}\\[0.3 cm] 4&=\dfrac{1}{2}\times k \times (9)^{2}\\[0.3cm] 4&=\dfrac{1}{2}\times k \times81\\[0.3 cm] k&=0.0987\ \text{N/cm} \end{align} $$


(a) Work needed to strech the spring from 41 to 43 cm

$$\begin{align} W&=\dfrac{1}{2}\times k \times (\Delta x)^{2}\\[0.3 cm] &=\dfrac{1}{2}\times 0.0987 \times (43-41)^{2}\\[0.3cm] &=\dfrac{1}{2}\times k \times81\\[0.3 cm] &=\boxed{\color{blue}{0.7896\ \text{J}}} \end{align} $$

(b) Distance beyond the natural length that a force of 40 N keep the spring streched

By using Hooke's Law

$$\begin{align} F&=kx\\[0.3 cm] 40&=0.0987\times x\\[0.3 cm] x&=\boxed{\color{blue}{405.2\ \text{cm}}} \end{align} $$


Learn more about this topic:

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

from

Chapter 4 / Lesson 19
201K

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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