**(a) the spring constant.**

**(b) the mass.**

A **spring-mass** system in straightforward terms can be **defined** as a spring system **where** a block is **suspended** or connected at the free end of the **spring.** The spring-mass system is **mostly** used to find the time of any **object** executing the simple **harmonic motion.** The spring-mass system can also be **utilized** in a broad variety of **applications.** For example, a spring-mass system can be **operated** to simulate the **movement** of human tendons utilizing computer **graphics** and also in the foot skin **deformation.**

Suppose a **spring** with **mass** $m$ and with spring **constant** $k$, in a sealed **environment** spring shows a simple **harmonic** motion.

\[ T=2\pi \sqrt{\dfrac{m}{k}} \]

From the **overhead** equation, it is **obvious** that the period of **oscillation** is unrestricted by both **gravitational acceleration** and **amplitude.** Also, a regular force cannot **change** the period of oscillation. The time **span** is directly proportional to the **mass** of the body that is attached to the **spring.** It will oscillate more **slowly** when a heavy object is **hooked** to it.

In **physics,** work is the **criterion** of energy **transfer** that happens when an object is driven over a **distance** by an outward force the **smallest** part of which is **applied** in the path of the **displacement.** If the force is steady, **work** may be **calculated** by multiplying the **length** of the **pathway** by the part of the **force** acting along the **way.** To describe this **idea** mathematically, the **work** $W$ is equivalent to the **force** $f$ times the **distance** $d$, that is $W=fd$. The work done is $W=fd \cos \theta$ when the force is **existing** at an angle $\theta$ to the **displacement.** Work **done** on a body is also **achieved,** for example, by **squeezing** a gas, spinning a **shaft,** and even by **compelling** invisible motions of the **particles** inside the body by an **exterior** magnetic force.

**A****cceleration, **in mechanics, is the **urgency** of change in the velocity of an **object** with regard to time. **Acceleration** is a vector quantity having **magnitude** and direction. The exposure of an object’s **acceleration** is presented by the **direction** of the net force operating on that object. Object’s **acceleration** magnitude is represented by **Newton’s** Second Law. Acceleration has the **SI** unit **meter** per second **squared** $m.s^{-2}$

## Expert Answer

**Part a**

The **formula** of work is given by:

\[ work = \dfrac{1} {2} kx^2 \]

**Rearranging:**

\[ k =2* \dfrac{work}{x^2} \]

**Inserting** the values:

\[ k =2* \dfrac{3.0} {(0.12)^2} \]

\[ k =416.67 \]

**Part b**

Two **different** formulas of **force** $f$ are given as:

\[ F =ma \]

\[ F =kx \]

\[ ma= kx\]

\[m = \dfrac{kx}{a}\]

**Inserting** the values:

\[m = \dfrac{(416.67)(0.12)}{15}\]

\[m = 3.33 kg\]

## Numerical Answer

**Part a: **$k = 416.67 N/m$

**Part b: **$m = 3.33$

## Example

Find the **period** of the spring given that it has a mass of $0.1 kg$ and a spring constant of $18$.

The **formula** to calculate the time period is:

\[T=2\pi \sqrt{\dfrac{m}{k}}\]

**Inserting** the values:

\[T=2\pi \sqrt{\dfrac{0.1}{18}}\]

\[T=0.486\]