John Doe
Physics Lab 161 202
Measurement Uncertainties Lab #1
Partners: Jack Smart
Felix D’cat
Computer Person: Jack Smart
Date: 01/30/2020
Instructor: Dr. Macoy
Introduction:
The purpose of this lab is to gain experience using uncertainties and to understand what
these uncertainties tell us about our measured values and the experiment. We will do this
by measuring the mass and diameter of a steel sphere. Using the equations,
ρ = m/V
V = (4пr^3)/3 = (п/6)d^3,
calculations for the density of the sphere will be performed. Where ρ is the density in
(kg/m3), m is the mass in (kg) and V is the volume of the sphere in (m3). The d and r
stand for the diameter and the radius of the sphere respectfully both in (m). We will also
be calculating the uncertainty in the diameter in order to find the uncertainty in the
volume then finally in the density. Finally, the experimental value of ρ will be compared
to the theoretical density of 7.8*10^3 kg/m3.
Procedure:
The first part of the lab we will use a meter stick to measure the diameter of the steel ball
ten different times. Then we use this value to calculate and find the uncertainty in the
volume. In order to find the density we will need the mass of the sphere. Using a scale,
with an error of .1%, the mass of the sphere will be measured. In the second part of the
lab a vernier caliper will be used to find the diameter of the sphere. Then we will
perform similar calculations to see if we get a better measurement than one with the
meter stick.
Steel Ball
Meter Stick
Vernier Caliper
Data:
Attached at the end of lab report.
Sample Calculation:
Volume using meter stick:
V = (п/6)*d^3 = (п/6)* (1.88cm +/- .07cm)^3 = (п/6)* (1.88cm +/- 3.7%)^3
%σV = (σV/(V))*100 = (.07/(1.88))*100 = 3.7%
%σV = 3 * %σV = 11.2%
Volume with uncertainty
V = 3.48 cm^3 +/- 11.2%
Density with uncertainty using meter stick:
ρ = m/V = (.03 kg +/- .1%)/(3.48 cm^3 +/- 11.2%)
%σρ = (.1%^2 + 11.2%^2)^(1/2) = 11.2%
ρ = .0086 kg/cm^3 +/- 11.2%
ρ = 8.6 * 10^3 kg/m^3 +/- 960 kg/m^3
Number of sigma away from accepted value:
N = (calculated – accepted) / uncertainty = (8600 – 7800)/960 = .8, within 1σ
Volume using vernier caliper:
V = (п/6)*d^3 = (п/6)* (1.91cm +/- .01cm)^3
V = 3.64 cm^3 +/- 1.5%
Density with uncertainty using vernier caliper:
ρ = m/V = (.03 kg +/- .1%)/(3.64 cm^3 +/- 1.5%)
ρ = 8.2 * 10^3 kg/m^3 +/- 123 kg/m^3
Number of sigma away from accepted value: 3.25, within 4σ
Conclusion:
This lab helped us learn how to make measurements using a caliper and how to use
measurement uncertainties to find the density of the steel ball and compare it with the
accepted value of 7.8*10^3 kg/m^3. Using these uncertainties we can find out whether
our measurements were accurate, precise, and realistic. In our trials we found the
average diameter of the ball to be 1.88 cm and the uncertainty in the diameter was .07 cm
using the meter stick. We took the average value because our measurements were
random. These values were calculated using excel. With these values we found the
volume of the ball to be 3.48 cm^3 with a percent error of 11.2 percent. The final
calculation of density turned out to be 8600 kg/m^3 with a percent error of 11.2 percent.
The calculated density for the steel ball ended up being within 1 sigma of the actual
value, giving us an a realistic measurement. Unlike the density value we found using the
vernier caliper, which turned out to be 8200 kg/m^3 and a percent error of 1.5 percent.
The density for the vernier caliper was with in 4σ from the actual value of density for the
steel ball. This told us that there was systematic error somewhere in the reading of the
caliper. No matter how many times the diameter of the ball was measured with the
caliper, we ended up with the same average value. The value with the vernier caliper was
a more accurate measurement than the meter stick because the percent difference was
smaller. It is also a more precise measurement because its sigma was smaller. This
showed us that just because the device is more accurate or precise does not mean that the
measurement will be a good one, because our value for the meter stick actually turned out
to be one sigma of the accepted value and the value for the caliper was with in 4σ away.
Making the value we found with the meter stick a better measurement. The sources of
error in this experiment were either the caliper was not calibrated correctly or the
measurement was not near the center of the ball.
Data
meter stick
diameter ruler
(cm)
1.8
1.9
1.9
1.9
1.8
1.8
1.9
1.8
2
2
Sum (cm)
Average (cm)
standard deviation
(cm)
Vernier Caliper (cm)
Mass (kg)
x-xbar
(x-xbar)^2
(cm)
(cm^2)
-0.08
0.0064
0.02
0.0004
0.02
0.0004
0.02
0.0004
-0.08
0.0064
-0.08
0.0064
0.02
0.0004
-0.08
0.0064
0.12
0.0144
0.12
0.0144
Sum
18.8 (cm^2)
1.88 RMS (cm)
0.078881064
1.91
0.0278
0.056
0.074833148