# A spring with a spring constant of 1.32 * 10^2 N/m is attached to a 2.0 kg mass and then set in...

## Question:

A spring with a spring constant of 1.32 * 10{eq}^2 {/eq} N/m is attached to a 2.0 kg mass and then set in motion.

(a) What is the period of the mass-spring system?

(b) What is the frequency of the vibration?

## Spring-Mass System:

Suppose we have a vertical spring-mass system setup, and when we commence the oscillation of the system, we can evidently notice that changing the mass of the suspended object greatly modifies the oscillation period of the system. Therefore, if we attach a heavier mass, the period of motion of the new system is undoubtedly longer than the period of oscillation of the original system. If we do not change the mass but have substituted an original spring with a stiff spring, the system will demand less time to accomplish an oscillation.

## Answer and Explanation: 1

**Given data:**

- {eq}k=\rm 1.32 \times 10^2 \ N/m {/eq} is the spring constant of the spring.

- {eq}m=\rm 2.0 \ kg {/eq} is the mass attached to the spring.

- {eq}T {/eq} is the period of oscillation.

- {eq}f {/eq} is the frequency of oscillation.

**Part a:**

The period of oscillation of the mass-spring system is:

{eq}\begin{align} T&=2\pi \sqrt{\dfrac{m}{k}} \\[0.3cm] &=\rm 2 \pi \times \sqrt{\dfrac{2.0 \ kg }{1.32 \times 10^2 \ \frac{N}{m}}} \\[0.3cm] &\simeq \color{blue}{\boxed { \rm 0.77 \ s}} \ \ \ \ \ \rm (correct \ to \ two \ significant \ figures ) \end{align} {/eq}

Therefore, the period of oscillation of the system is {eq}\color{blue}{\boxed { \rm 0.77 \ s}} {/eq}.

**Part b:**

The frequency of oscillation is given by:

{eq}\begin{align} \rm Frequency&=\rm \dfrac{1}{period} \\[0.3cm] f&=\dfrac{1}{T} \\[0.3cm] &=\rm \dfrac{1}{0.77 \ s} \\[0.3cm] &\simeq \color{blue}{\boxed { \rm 1.3 \ Hz}} \ \ \ \ \ \rm (correct \ to \ two \ significant \ figures ) \end{align} {/eq}

Therefore, the frequency of oscillation of the system is {eq}\color{blue}{\boxed { \rm 1.3 \ Hz}} {/eq}.

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Chapter 17 / Lesson 11In 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.