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## What to submit:

• A lab report is required.
• Submit one pdf of the Lab report
• You have one week to work on this lab.

## Lab Manual:

Simulation to be used for this Lab: masses-and-springs-PHET (Links to an external site.) the link for this

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## Instruction video:

Here is the video that will walk you through the Lab: Spring constant lab.mp4 Download Spring constant lab.mp4Play media comment.

Spring Constant and Oscillations Lab

1 Introduction 1.1 Hooke’s Law In the 17th-century British physicist Robert Hooke introduced a law that de-scribed the characteristic stiffness of a spring. Where the force of a spring would scale linearly to the distance that the spring is stretched, compressed. To no surprise, this law was named after him;”Hooke’s Law”.

Fs = −kspringx (1)

Figure 1: Spring Force in Action

The negative in the equation illustrates that the force for a spring goes in the direction opposite to the displacement. In this lab we will be measuring the

1

Figure 2: Mathematical Model of Simple Oscillation

spring constant. kspring. We may do so by observing the spring in oscillation

and equilibrium due to a hanging mass. We will be performing both experiments in this lab

1.2 Oscillations and Sinusoidal Functions

In your lectures you may have discussed about oscillations here is an overview: The mathematical specimen in figure 2 is a sinusoidal function and in this

case we are modeling our oscillation in the time domain. You may ask yourself why time? A simple explanation would be in the case of this model it is more intuitive and thus produce our following simpler derivations. Remember we have oscillations in both space and time! In your spare time look up wave numbers! We may observe some notable characteristics of the function, such as the period which is denoted by T and the amplitude A which represent the peak of the function, in our case the maximum displacement x, and the Force Fspring. If our system is oscillating with a constant period we are allowed to use this model for displacement. Note A = maximum displacement

x(t) = A cos( 2πt

T ) (2)

A in this case is the maximum displacement in this case. To simplify our derivation we will note one mathematical property of functions in general. If we have a function

f(x) = x2 (3)

h(x) = f(x) + C (4)

The function h(x) will be f(x) shifted on the Y axis by a constant C according to equation 4 and have no effect on the characteristics of the function. This applies

2

to our case because in the system of a hanging spring we have the following. Note we switched x to y because we are working vertically so we measure displacement where y is centered where the force of a spring is 0

Fnet = −ky − mg (5)

From the latter information stated, we may further simplify our derivation for the spring constant with little in depth knowledge of differential equations, if we observe a spring connected vertically we can write F = ma

Fnet = ma = −kx (6)

Fnet = m d2x(t)

dt2 (7)

We Know Hooke’s is the force described by a spring.

−kx(t) = m d2x(t)

dt2 (8)

Now we can use our generic model of oscillation and see what happens when we put it in place for x(t)

−kA cos( 2πt

T ) = −Am

4π2

T2 cos(

2πt

T ) (9)

To further simplify and familiarize what you have already probably seen in your text books. we will declare anew characteristic called the natural frequency

ω = 2π

T (10)

After simplification we may write the following.

k = mω2 (11)

Thus we may solve for omega √ k

m = ω (12)

Which we may relate to T by plugging this back in.√ m

k 2π = T (13)

Interestingly enough, the simple mathematical model we assumed for displace- ment to be true, results in a fascinating description for characteristic oscillation of a spring. It can be safe to state that it is completely independent of the amplitude. So no matter how much we pull back the spring, it will oscillate at the same frequency! A natural frequency dependent on the construction of from basic characteristic properties of the system that will remain unaltered by through pure kinematics. Altering the components of the system will change its characteristics, such as stretching the spring so far as to a point to where we loosen the spring or simply changing the mass, however doing so changes the characteristic frequency of the system .

3

2 Apparatus • PhET simulation:

• 3 spring constants (Small, Middle and Large)

• Masses

• Ruler

• Damping (None)

4

3 Setup

5

4 Measuring kspring with Displacement(Space)

In This section we will observe the spring in equilibrium, meaning that

(14)

(15)

Fnet = 0

Fnet = kspringx − mg

kspringx = mg (16)

This equation will allow us to calculate the force that the spring that is applied on the spring.

1. Use spring with spring constants (Small)

2. Place a mass on the lower end of the spring and weight and allow it to stretch into equilibrium

3. Take your ruler and measure the length of the stretched spring. Record the length.

4. For this instance calculate the force of gravity.

5. Do this for 10 increasing weights, and we should now have a measurement of 10 Forces and 10 lengths. Note: Add weights so we notice an observable increase in the length of the spring

6. Graph the length in meters vs the Force in Newton, we should observe a linear plot.

7. Take the linear regression of the plot and the slope should be our mea- surement for kspring (smaall)

8. Repeat experiment for springs with spring constants (Middle and Large)

5 Measuring kspring with Time

1. Use spring with spring constants (Small)

2. Hang a mass on the lower end of the spring and pull back the mass to begin an oscillation.

3. Record the time period on stopwatch for 10 oscillations and then divide it with 10 to get one time period.

4. Solve for kspring from equation 13

5. Repeat this experiment for for springs with spring constants (Middle and Large)

6

6 Comparing kspring from both the methods

1. Find the percentage error in kspring (small, Medium and Large) from both the methods (4 and 5).

7 Find unknown masses

• Using the known value of kspring (Large) find the value of known mass(blue) using both methods (4 and 5)

9 Conclusion and Key Concepts

• Why does the spring Oscillate? And why can you also have it at equilib-rium

• In real world, there is no such thing as perpetual motion, so why does the springs am-plitude begin to dampen? Where does this energy go?

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