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Objectives

In this experiment we study the physical system composed by a spring and

a mass attached to it. The main goals are to check the validity of Hooke’s

law, which we believe is a good model for this system, and to understand the

oscillatory movement that springs undergo when taken initially away from

their equilibrium position.

Theory

A spring is a mechanical device that can compress or expand when a force

is applied to it, but it returns to its original position when the force is

removed. In particular, if the relation between the force applied (Fappl) and

the expansion/compression (x) is linear, as in

Fappl = kx (1)

the system is especially simple. By using Newton’s third law, if there is an

external force Fappl acting on the spring, the spring is acting with an equal

and opposite force Fs = −Fappl. Then, the force exerted by the spring is

given by

Fs = −kx. (2)

This is called Hooke’s law, and it appropriately describes the behavior of

real springs if the forces applied are small. The behavior of a spring under

this model depends only on the value of k, which is the spring constant.

Equilibrium

The equilibrium position for a spring happens when the force is equal to

zero. This implies that x = 0 is the equilibrium position, which is why x is

usually called displacement from equilibrium. In a situation where there are

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several forces acting on the spring, the equilibrium position depends on all

of them. For example, if we attach a mass to the end of a spring and hang it

vertically, the equilibrium position corresponds to the point where the force

of the spring compensates the force of gravity. In this case, the total force

must cancel:

Fs + Fg = 0

Fs = −Fg,

−kx = −mg,

x =

mg

k

.

(3)

Keep in mind that in this manual, x is positive if it points down, and negative

if it points up, with respect to the equilibrium position.

Fg

Fs

equilibrium

Figure 1: Mass hanging from a spring.

Oscillations

The nature of Hooke’s law makes springs show a peculiar behavior when

placed away from their equilibrium position. When the object in the picture is slightly pulled down, away from its equilibrium position, the spring

force will pull upwards. This force will overcome gravity and produce a net

displacement upwards. As the mass moves up, it will pass the equilibrium

position to lay above it. In this situation, the spring will be compressed,

and its force will push down. At some point, this force will make the spring

expand, moving the mass down, past the equilibrium position, and going

back to the starting place, where the object is below the equilibrium position. This process will automatically repeat periodically, and the movement

produced by it is what we call simple harmonic motion.

2

The oscillations in the spring will repeat periodically. This means that

the position and velocity of the object will be the same at some time t and

at a later time t + T, a later time t + 2T, a later time t + 3T, etc. The

quantity T is called the period of the oscillations, and for a spring driven by

Hooke’s law is given by

T = 2π

r

m

k

, (4)

which can be otherwise stated as

T

2 =

4π

2

k

× m. (5)

Online experiment

The online simulation for this experiment can be accessed at

https://phet.colorado.edu/sims/html/masses-and-springs/latest/massesand-springs en.html

Once there, enter the Lab option:

• on the left, you will see an energy graph which will show how much

kinetic, gravitational, elastic and thermal energy the spring has at a

given moment;

• on top, you will see the spring, and two selectors: one for the value of

a mass, and another for the value of the spring constant (k);

• at the bottom, you will see three masses. The orange has its mass

(m) selected by the slider described above. The other two have, in

principle, unknown values;

• on the right, you will see several boxes that enable showing some

reference lines in the picture (displacement, equilibrium position, …),

a slider for the value of gravity (which you don’t need to use), and a

slider to select damping. You can also see a ruler and a stopwatch,

which you will need to use to perform measurements.

Procedure

Part A: determining k using Hooke’s law

1. Choose a value for the spring constant k at random using the slider at

the top. Although not impossible, part B will be harder to do if k is

3

too high.

2. Move the slider for damping all the way to the right (Lots).

3. Check the first and second boxes on the right panel to see the natural

length of the spring, and the position of the masses.

4. Choose a mass for the orange object of m = 50 g, and hang it from

the spring.

5. Using the ruler, measure the distance between the blue and the black

lines.

6. Record the current value of the mass and the distance the first and

third columns of the table.

7. Repeat steps 4 to 6 for a total of six values of m between 50 g and

300 g.

m (g) m (kg) Fg (N) x (cm) x (m)

Table for part A.

Part B: determining k using the period of oscillations

1. Keep the value of k from part A.

2. Move the slider for damping all the way to the left (None).

3. Un-check all boxes on the right panel and put the ruler away, as none

of those are needed in this part.

4. Chose a mass for the orange object of m = 50 g, and hang it from the

spring.

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5. Slightly pull the mass away from its equilibrium position.

6. Start the stopwatch when the mass is exactly at the bottom of its

movement.

7. Count the number of times it reaches the bottom after that (not including the first one), and stop the stopwatch exactly as it reaches the

bottom for the tenth time.

8. Record the current value of the mass and the total time indicated by

the stopwatch (ts) on the first and third columns of the table.

9. Repeat steps 4 to 8 for a total of six values of m between 50 g and

300 g.

m (g) m (kg) ts (s) T (s) T

2

(s2

)

Table for part B.

Calculations and analysis

Part A

1. Convert the masses (m) from g to kg, and the positions(x) from cm

to m, filling the second and fifth columns of the table.

2. Calculate the gravitational force (third column) as

Fg = mg.

3. On a spreadsheet software, make a graph with the force Fg in the

vertical axis and the distance x in the horizontal axis.

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4. Find the slope and the intercept of the best fit straight line. A general

straight line is

y = ax + b,

where a is the slope and b the intercept. Comparing to the equilibrium

equation

Fg = kx,

the slope of the line should be the constant of the spring: k = a. Give

the value of k in the appropriate units.

5. What value of the intercept do you obtain? How does this compare

with the value you expected to obtain?

Part B

1. Convert the masses (m) from g to kg, filling the second and fifth

columns of the table.

2. Calculate the period (fourth column) from the stopwatch time (third

column) as

T =

ts

10

.

Calculate the square of the period, to fill the fifth column.

3. On a spreadsheet software, make a graph with the squared period T

2

in the vertical axis and the mass m in the horizontal axis.

4. Find the slope and the intercept of the best fit straight line. Comparing

a general straight line with equation (5) the slope (a) is related to the

spring constant as

a =

4π

2

k

→ k =

4π

2

a

.

Give the value of k in the appropriate units.

5. Compare the values of the spring constant obtained in part A (kA) and

part B (kB). To do so, calculate their percent different with respect

to their average value:

% diff = 100% ×

kA − kB

kA+kB

2

.

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