 # A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. 3.0 J of work is required to compress the spring by 0.12 m. If the mass is released s from rest with the spring compressed, it experiences a maximum acceleration of 15 m/s^2 . Find the value of (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.

Acceleration, 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 $m.s^{-2}$

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:

$m = \dfrac{(416.67)(0.12)}{15}$

$m = 3.33 kg$

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:

$T=2\pi \sqrt{\dfrac{0.1}{18}}$

$T=0.486$