What is an inertial balance?
“Inertial mass is measured with the use of an inertial balance, or springloaded pan. It is a dynamic measurement – that is, a measurement that can only be accurately recorded while the system is in a state of motion.
This method capitalizes on an object’s inertia, or its tendency to continue in its current state of motion, as a means of quantifying the amount of matter present.” (PhysicsLab Online)
We’re going to add masses to this pan, set it in motion, and then use a photogate to measure the resulting period.
The period is the time taken to repeat the pattern of motion once. This kind of motion is called an oscillation, or simple harmonic motion.
Here, the period (T) is the time it takes for the mass to go from the bottom, top, and then back to the bottom again.
This setup will be placed so that part of this oscillating pan will pass through a photogate, as shown below.
So what is a photogate?
An inertial balance is two metal pans connected by steel strips. We clamp one end to a table, and the other end is free to oscillate. As we add mass to the balance, it’s inertia increases, and so oscillates more slowly.
“By comparing the periods of the apparatus with known masses to the period of an unknown mass, the value of the unknown mass can be determined. This comparison can either be accomplished using a graphical program such as Excel or graphing the values by hand. … provides a definitive way to determine the inertial mass of an object directly, without relying on gravity.” (Arbor Science)
Pan, Cclamp, tabletop, and CPO timer.
__________________________________________________________
Purpose:
* Learn how to find an object’s mass, without a need for gravity! (Scales only work if gravity pulls a mass downwards – so scales won’t work in zerog situations.
Use data to produce a calibration curve and find the unknown mass of an object.
Learn how to make LoggerPro perform calculations for us
Learn how to precisely interpolate an answer from between data points
Equipment:
Inertial Balance, large Cclamp
Collection of known slotted masses
Masking Tape
Calculator
CPO Timers, and CPO photogate, small Cclamps
Unknown Mass – get from teacher after completing data table
Procedure

Record the mass of your inertial balance (150 g)

Clamp the inertial balance to the counter (not the table)

Add a mass to the pan (make sure that it is secured in place)

Tape cardstock to the outer edge of the pan. Set it up so that when it moves, the card will trigger the photogate as it passes through.

Set the CPO timer function to period.

Record the period of the pan oscillating.
Don’t use the first reading – rather, take the 2 or 3^{rd} . But take data before the oscillation slows down.

Now add more mass, set pan in motion. Record the new period and the mass in the table below.

Repeat adding masses until you have 8 data points.
Use a wide range of mass ( 0 – 300 g) to get the best data.

Get the unknown mass and find it’s period.

Get the accepted value for the mass on the electronic balance
Be careful that the masses do not slide around in the pan as this causes damping.
Data Table:
Mass of balance = _______________ g
½ mass of balance = ______________ g
Mass, Period & Period^{2}
Mass added to pan  Total Mass (g) in motion 
Period (seconds)  Period^{2} (seconds^{2}) 
20 g  20+ ½ mass of inert bal  __  __ 
__  __  __  __ 
__  __  __  __ 
__  __  __  __ 
__  __  __  __ 
__  __  __  __ 
__  __  __  __ 
Unknown Mass : Data
Describe your unknown mass 

Period (s) 

Measured value of mass from electronic balance (g) 
Analysis

Use Logger pro. In the Xaxis plot [grams + 1/2 mass of balance]
in the Yaxis plot Period. 
‘Data’ > ‘New calculated column’

Name this new column “Period^2”, units “s^2”

Under Equation, look for the “Variables (columns) >” box. Click, choose “Period^2”
Choose “Period” then *,then “Period”
Logger pro will then automatically create a new column for us.

Click the Yaxis where it says ‘Period’; change it to ‘Period^2’.

Autoscale button

Use the “R=” button to produce a linear best fit. Make sure that all data points are included in this regression analysis.

Write down the equation for the lines

Now we’ll use a process called interpolation to find the mass of the unknown object we measure: Under the Analyze tab, click Interpolate.

Move the interpolation line until it’s period matches the period of our unknown: Now read off the corresponding mass!

How did we do? Let’s compare our interpolated result with the actual mass measurement of the “unknown” mass, which we get from the digital scale. calculate the percent error:
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Questions : Use complete sentences

Describe the motion of the pan. Will the pan ever be at rest? Explain.

Would an inertial balance work in a zero gravity environment? Explain.

Would a spring balance (like we used for the mass/weight lab) work in a zero gravity environment? Explain.

Are mass and weight identical? Are mass and inertia equivalent? Discuss.
What is inertia? What is mass?
How is mass related to weight?
Inertia, mass and weight
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Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HSPS21. Analyze data to support the claim that Newton’s second law of motion is a
mathematical model describing change in motion (the acceleration) of objects when
acted on by a net force.
HSPS210(MA). Use freebody force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations
A FRAMEWORK FOR K12 SCIENCE EDUCATION: Practices, Crosscutting Concepts, and Core Ideas
PS2.A: FORCES AND MOTION
How can one predict an object’s continued motion, changes in motion, or stability?
Interactions of an object with another object can be explained and predicted using the concept of forces, which can cause a change in motion of one or both of the interacting objects… At the macroscale, the motion of an object subject to forces is governed by Newton’s second law of motion… An understanding of the forces between objects is important for describing how their motions change, as well as for predicting stability or instability in systems at any scale.
Massachusetts Science and Technology/Engineering Curriculum Framework (2006)
1. Motion and Forces. Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.4 Interpret and apply Newton’s three laws of motion.
1.5 Use a freebody force diagram to show forces acting on a system consisting of a pair of
interacting objects. For a diagram with only colinear forces, determine the net force acting on a system and between the objects.
1.6 Distinguish qualitatively between static and kinetic friction, and describe their effects on the motion of objects.
1.7 Describe Newton’s law of universal gravitation in terms of the attraction between two objects, their masses, and the distance between them.
1.8 Describe conceptually the forces involved in circular motion.