PROJECT

Purpose

You will investigate the mechanical system of a spring with an object in this project. The motion of the weight is govened by a physical law (Hooke’s Law) and its displacement in time can be computed from Newton’s second law. The law establishes a second order linear ODE with constant coefficients for the displacement of an object. You are expected to understand the mechanism of the system to accomplish the goal of this project. Please read the following problem description carefully to answer the questions.

Project Report Format

Project report should have four main sections including (i)Introduction including the purpose or aim of project, (ii)Materials and Methods, (iii)Results and Discussion.

Problem Description

Consider an object hanging from a spring shown in Figure 1. The weight w of the object Figure 1: A Simple Mass-Spring System

is the magnitude of the force of gravity acting on it. The mass m is is related to w by w = mg where g is the gravitational acceleration g = 9.8m/s2

. Assume that the spring obeys Hooke’s Law: Force is proportional to displacement. It means that if the spring is stretched or compressed a distance s from its neutral, or unstretched, length, then the magnitude of the force F exerted on the spring is given by the formula F = ks (1)

where k is called the ’spring constant’, which relies upon purely the mechanical properties

of a spring. Figure 1a and 1b show Unloaded spring and Loaded spring in equilibrium. The

Hooke’s law states that by taking w as a force

w = kso (2)

where so is the displacement by the weight of the object (Weight Force), w. We now define y

as the displacement upward from the center position of the object. Therefore Figure 1b and

1c illustrate that if the spring is stretched, the displacement of the spring becomes so − y,

which is positive. In other words, if the spring is compressed, so − y will become negative

because y is larger than so from Figure 1c and 1d. Because so − y is the displacement of the

spring, the Hooke’s law states that the spring force due to the stretch or compression of an

object would be

F = k(so − y) (3)

and the force is called Elastic Force and its direction is upward.

Consider the Newton’s second law for an object,

XF = ma = m

d

2

y

dt2

(4)

which means that the acceleration a of an object is determined by forces exerted on the

object. Forces exerted on the the center of the object streching and compressing (Vibrating)

are (i)weight force, Equation 2, whose direction is downward and (ii)spring force (vibrating),

whose direction is upward. Hence, the total force in the direction of y is

XF = k(so − y) − w = −ky (5)

Then Equation 4 becomes

m

d

2

y

dt2

+ ky = 0 (6)

This equation is called an equation of harmonic motion.

A realistic analysis of the vertical motion of the mass however would take into account not

only the elastic and gravitational forces but also the effects of friction and all other forces

acting externally on the object. The frictional force due to the friction of the spring is

proportional to the velocity of the object and thus,

F rictional F orce = −c

dy

dt (7)

where c is the propotional constant and the minus sign indicating that the resistance always

acts in opposition to the velocity. The total forces including the frictional force become

XF = k(so − y) − w − c

dy

dt = −ky − c

dy

dt (8)

Then Equation 4 becomes

m

d

2

y

dt2

+ c

dy

dt + ky = 0 (9)

The motion of the object is characterized into three catagories by the following conditions:

c

2 − 4mk > 0, or = 0 or < 0 (10)

If Equation 10 > 0, the motion is said to be overdamped. If it is less than 0, then the motion

is underdamped. Finally if it is equal to zero, the motion is said to be critically damped.

Based on the theory, you will perform a mini-experiment for a Mass-Spring System to answer

the questions below. Go to the website, https://phet.colorado.edu/sims/html/massesand-springs/latest/masses-and-springs en.html to do your experiment. Do not copy

the address and paste it to open the page. Click the link provided in the Canvas announcement.

Animation Instruction

On the webpage, click ’Lab’ and you will see a spring attached to the top wall. There are

cylinders that can haning from a spring by clicking and holding a cylinder to place it to the

bottom of the spring. You can change the mass using the top left control box and change

spring constant, k using the top right control box. On your right, there is a grey colored

control box, which you can click ’Mass Equilibrium’ to show a dashed line, the equilibrium

position. Set Gravity to 9.8m/s2

for Earth. You can change the value of Damping from None

to Lots in the box. Below the box, there are a ruler that you can measure the displacement

’y’ and a stop watch to measure time. On the bottom right, you can find control switches to

stop or resume or slow your animations. The top pink circle button right next to the spring

wall can set the position of cylinder to the original equilibrium position.

Answer the following questions.

(a) Describe Equation 9, giving its order and telling whether it is linear or nonlinear and

homogeneous or non- homogeneous. State the reasons,

(b) You will test Hooke’s law using Equation 2. Change the mass of your object and spring

constant k and create several graphs for the weight w v.s. the equilibrium displacement

s for different spring constants k. Do your results obey the Hooke’s law?

(c) Consider Equation 9 and choose a spring constant and mass. In the animation, change

”Damping” from None to Lots to see the behavior of the motion of the object. Use

the ruler to plot the curves of the displacement of the object y v.s. time t for several

damping cases by labeling the damping cases as the numbers from 1,

(d) Find the analytic solution of Equation 9 in terms of m, k, and c. Determine the conditions of c for (i)overdamped, (ii) underdamped, and (iii)critically damped for your choice

of m, k. Note that three conditions will result in three different analytic soltutions.

(e) Approximate the actual values of c from the curves of your experiment by comparing

the plots from the analytic solutions. You can use any graphing softwares to roughly

estimate the values by your eyes or other sources. What are the values of the analytic

c and the label from the experiment for ”critically damped” case?

.

So far we have considered free damping examples without a constant external force. Applying

an external force to the free motion system results in the system of Forced Motion. For

example, we can consider a sinusoidal or cosinusoidal impressed force as an external force.

Then the differential equation 9 becomes

m

d

2

y

dt2

+ c

dy

dt + ky = Fo cos(ωt) (11)

where Fo is the amplitude and ω is the angular frequency, respectively, which determine the

characteristics of the cosinusoidal external source. Answer the following questions.

(a) From the analytic solution with your chosen m, k for critically damped c, find your two

values of ω, which satisfy the following conditions:

1. ω is not equal to the roots of the characteristic equation for homogeneous solutions.

Choose any value satisfying this condition.

2. ω is equal to the roots.

(b) Find the analytic solutions for the above two cases with your choice of Fo and plot them and describe the motions.

(c) Discuss the differences between Free motion and Forced motion based on your results.