How to calculate uncertainty of spring constant

1 PHY 241 Fall 2018

PHY 241 Lab 5- Determining a spring constant

Introduction:

Our goal for this week is to learn how to use the CBR data to

extract information about parts of our system. Specifically, we are

going to measure the spring constant of a single spring through two

different methods.

The system we are interested in this week can be drawn

schematically as shown to the right. To find the spring constant,

we will start take advantage of Newton’s Second Law and Hooke’s

Law. Unfortunately, because the springs we will be using have

coils which are touching each other when the spring is not under

tension, the Hooke’s Law equation

|𝐹𝑠𝑝𝑟𝑖𝑛𝑔| = 𝑘 |Δ𝑥|

is only approximately correct.

Equipment:

CBR 2- connected directly to a computer using a USB cable

Spring

Cart

String with loop knots in each end

Set of Masses

Hook

Paperclip

Pulley

2 m track

Bubble level

Computer with Logger Pro or Logger Lite and Excel.

Procedure:

1) Arrange your equipment to match Figure 1. Figure 2 shows a fast and easy way to connect the different components. Make sure you place the CBR in a location where it can measure

either the cart or the hanging mass as it moves.

spring

𝑚2

𝑋

𝑚1 Pulley

Figure 1

H

2 PHY 241 Fall 2018

2) Group- Compare the random uncertainty of the CBR to a ruler/tape measure. 3) Group- Compare the systematic uncertainty of the CBR and the ruler/tape measure.

4) Researcher- Considering the previous two answers about uncertainty, why are we using the CBR for Task 1 and Task 2?

Task 1

5) Using a variety of masses on the hook, measure the position of the cart (or hanging mass) while it is motionless using the CBR. Keep the maximum amount of mass hanging from the

string LESS than 600 g.

6) For each trial use Excel to determine the stretch of the spring, Δ𝑥, and the force, 𝐹 that spring must have been exerting to keep the cart motionless.

7) DA- Generate a plot to compare the force of the spring to the stretch of the spring. Since we are using 𝐹𝑠𝑝𝑟𝑖𝑛𝑔 = 𝑘Δ𝑥 as our guide, make sure that the stretch is on the horizontal

axis.

8) Notice that a linear trendline (complete with equation) should neatly fit your data.

9) PI- What does it mean that the trendline does not pass through the origin? (Note: the y- intercept should not be close to the origin).

Task 2

10) Place a large amount of mass on the hook so that the car can bounce back and forth a few times after being pushed. (Note: If the spring bangs against the track during a data run you

should retake that data.) Also set your sampling rate high (between 30 and 40 samples per

second) so you are taking many data points for each bounce.

11) Import all the data from Logger Pro into Excel.

12) DA- Generate two plots: One of stretch vs velocity and the other of stretch vs acceleration. Feel free to use calculated columns to clean up your data before plotting. (For example, the

instructor has provided a formula which corrects for friction based on Lab 3.)

paperclip

Figure 2

3 PHY 241 Fall 2018

a. PI- One of these plots is linear and needs to be explained quantitatively. You

should be able to use Newton’s second law to connect the slope and y-intercept to

the spring constant, 𝑘, and the masses involved.

i. Don’t forget to find numbers for both the random and systematic

uncertainties in your work.

b. DA- One of these plots is not linear and needs to be explained qualitatively. You do

not need to calculate numbers, but you do need to explain the patterns that are

appearing in the data. Here are a few questions to help you identify different

patterns.

i. Is there any meaning to the general shape of the data?

ii. Is there any meaning to the average value of the data, either vertically, �̅�, or

horizontally, �̅�?

iii. Is there any meaning to the spread of the data, either vertically, 𝛿𝑦, or

horizontally, 𝛿𝑥?

13) PI- Do your results for Task 1 and Task 2 agree with each other? If not how large of a systematic uncertainty exists between your two methods? If possible, it is strongly

recommended that you take the Task 2 data a few times to make certain your results are

consistent.

14) Before leaving the classroom, make sure you email the data out to the entire group and clean up your work station, returning small equipment to the appropriate storage.