Measuring Mass without a Scale: The Inertial Balance

Purpose:

To use an inertial balance to measure mass. First, you will "calibrate" the balance using known masses, then use the balance to find the mass of "unknown" objects.

Discussion:

"Mass: The quantity of matter in a body. More specifically, it is a measure of the inertia or "laziness" that a body exhibits in response to any effort made to start it, stop it, or change in any way its state of motion."

(Hewitt, Paul, Conceptual Physics, Second Edition, 1992, p. 32)

Scientists measure things. A scientific question to ask is "This definition of mass is very nice, but what does it say about measuring mass?" There are several ways to measure mass - a triple-beam (or electronic) balance measures mass, for instance. The triple-beam balance has a couple of disadvantages, however. First, it is difficult to see how the measurement you make on a balance correlates to the definition of mass given above, and the triple-beam balance won't work where there is no gravity. In this lab we will measure mass by utilizing its true nature, that of resisting any change in its state of motion.

If mass measures the "laziness" of an object in response to efforts made to change its velocity, it makes sense that you should be able to measure mass by making an effort to change the velocity of an object and recording its "laziness". This is what an inertial balance does. Two strips of spring steel apply a constant amount of "effort" in order to vibrate a pan back and forth. (A vibration involves speeding up, slowing down, and changing direction (all 3 ways to accelerate), so the state of motion of the object is certainly changed.) If the object can be vibrated back and forth easily, it is not "lazy" - in other words, it does not have much mass. Objects that vibrate slowly have a large mass.

By measuring how fast known masses vibrate on the inertial balance, you can construct a graph that "calibrates" the balance. By determining the mathematical relationship between the mass of the object and its period of vibration, you can calculate the mass of your unknown objects.

Equipment:

 inertial balance graph paper or Excel stopwatch masking tape (to hold masses) C-clamp (to attach balance to table) masses

Procedure:

NOTE: You will work with one or more lab partners in this lab. You are responsible for turning in INDIVIDUAL lab reports, however. Your lab report should include a data table, your graph, results for the "unknowns", and analysis.

Part 1 - Calibrating the Balance

The instructor will demonstrate how to set up the inertial balance. Be sure to clamp one end of the balance to the table so that the other end can vibrate freely in the air beside the table. It is usually easier to clamp the balance under the edge of the table instead of on top of it. When you place objects in the balance pan, you will need to use small pieces of masking tape to keep them from sliding about in the pan.

The object of calibrating the inertial balance is to come up with a graph that shows the response of the balance when a range of masses is placed in it. To do this, you will need to do some careful planning. Here are some hints and pointers:

• You will need to use as wide a range of masses as practical - from 0 grams up to as much mass as the inertial balance will hold without buckling. Most of the balances we have will hold 500 - 600 grams. I recommend changing the mass by about 50 grams per trial. The "heavy duty" models will hold more, and you might want to increase mass by 100 grams or so per trial. Don't worry, you won't get too much data...You don't have to take the masses in strict order - you can come back and fill in "gaps" in your data.
• You can determine the response of the inertial balance by measuring its period - the time it takes for one complete vibration (over and back). Actually, you don't need the period itself - the time for 10 vibrations (10 periods) is easier to work with.
• Use a triple-beam balance or an electronic balance to find the masses of each of the knowns.
• Don't try to time one period! Record the time for 10 vibrations for each trial.
• Perform three trials of each mass to catch timing or counting mistakes.
• Construct a data table in Excel similar to this one to record your data.
Item #
Mass
(g)
Mass
(kg)
Time for 10 vibrations (s)
Trial 1
Trial 2
Trial 3
Average
1

2

3

4

5

6

7

8

9

first unknown
# ____
don't include this data in your graph - include the masses of the unknowns in your analysis and conclusion

second unknown
#____

• Graph your data (not including your unknowns) as a scatterplot of the time for 10 periods (y-axis) v. mass (x-axis).

Determine the best-fit straight line and display its equation and R2 value.

You will use this equation to determine the masses of your unknowns.

Part 2 - Measuring "Unknown" Masses

You need to demonstrate that you can measure the mass of an object using the inertial balance. Your instructor will place several objects of "unknown mass" where you have access to them. Determine the mass of 2 (two) of them using your inertial balance and the equation of the best-fit line from the time v mass graph.

• Example: Suppose that the best-fit line for your data is y = 5.0x + 10. On your graph, the y-axis represents the time for 10 vibrations, and the x-axis shows the mass.
• Solve your equation for x, the mass:
• y - 10 = 5.0x
• (y - 10)/5.0 = x
• Let the time for 10 vibrations of your first unknown = y;
in this case, your first unknown took 14.0 seconds to vibrate 10 times.
• Calculate x.
• (14 - 10)/5.0 = x
• 4/5.0 = x
• x = 0.8 kg
• Be sure to record your measurements for the unknown masses in a data table (the same one will work) and be sure to identify each unknown by its number.
• IF instructed: Measure the mass of the unknown in a more conventional manner - using a triple-beam or electronic balance, for instance. This provides a check on the accuracy of the inertial balance.

Results

Use the equation of the best-fit line from your graph to determine the masses of your unknowns.

Analysis:

1. What are some advantages of timing 10 vibrations of the inertial balance instead of just one?
2. How accurately does the inertial balance measure the masses of your unknowns? What limits its accuracy? (Be specific, and support your answer; "human error" is not acceptable because it can be corrected.)
3. Would the inertial balance successfully measure mass in the Space Shuttle when it is in orbit around the Earth? Why do you think so? What about a triple-beam an electronic balance, which is the more-common way of measuring mass on Earth?
4. Using an electronic balance, measure the mass of the unknown and calculate the percent difference between the two values. (Your teacher has found this actual mass and will share it with you.)
5. Which value, from the inertial balance or electronic balance, measures the fundamental nature of matter? Why do you think so? (HINT: Read the first paragraph of this lab!)
6. Which value, from the inertial balance or electronic balance, do you believe is more accurate? Why do you think so?
7. Why is the inertial balance set up so that it vibrates horizontally instead of vertically?

(modified from J. Stanbrough @ www.batesville.k12.in.us)

Unknown #
Mass (g)
1
333.2
2
332.7
3
331.7
4
332.4
5
331.9
6
332.8
7
332.3
8
331.9
9
332.6
10
331.6
11
331.6
12
NA
13
NA

An electronic balance determines mass by measuring how hard it's pushing up against the mass.