PHYSICS LAB: THE INERTIAL BALANCE



PHYSICS LAB: THE INERTIAL BALANCE (revised 9/11)

Introduction

The mass of a body is typically defined as the quantity of matter contained within the volume of that body. Measuring that quantity of matter either involves measuring the volume and calculating from a known density or using a laboratory balance to measure the mass directly. This direct method uses the fact that the weight of an object is directly proportional to its mass, provided the gravitational pull is constant. Unfortunately, students often refer to mass and weight interchangeably which leads to the misconception that they are the same quantity.

The purpose of this lab is to present an alternate method of determining an object’s mass which is independent of the gravitational pull thereby enhancing the student’s appreciation of the differences between mass and weight.

Theory

The mass of a body may exhibit itself is two ways:

1. As Inertial Mass, or the property of a body to resist changes in its state in motion. This property is described in Newton’s 1st law (sometimes referred to as the Law of Inertia).

It can be derived (using Calculus relationships) that by applying Newton’s 2nd law to the motion of an object in Simple Harmonic Motion the period of the oscillations is directly related to the mass of the object. More specifically,

(Period)2 ( Inertial Mass Eqn[1]

Where the symbol ( means the two quantities are directly proportional.

A graph of (Period)2 versus mass (mass on the x-axis) would yield a linear relationship with a slope and y-intercept. In equation form, this would be

(Period)2 = m ( Inertial Mass + b Eqn.[2]

where m is the slope of the graph and b is the y-intercept, which should be the period-squared value for a zero inertial mass.

2. As Gravitational Mass, exhibited by the mutual attraction of one mass for another as outlined in Newton’s law of Universal Gravitation. Near the earth’s surface where the gravitational field is constant, the gravitational mass is related to the object’s weight by

Weight = Gravitational Mass ( g Eqn.[3]

Where g is the acceleration due to gravity having a value of 9.81 m/s2 (or 32.2 ft/s2). Gravitational mass is usually measured using a spring scale or triple-beam laboratory balance.

Since mass is mass, the Inertial mass should be equivalent to the Gravitational mass. Or, measuring mass by two distinct methods should provide equivalent results.

Experiment

One way to determine the mass of an unknown is to compare its gravitational mass with standard masses using a laboratory balance. However, if one found oneself in the weightlessness of space, a laboratory balance would not operate in zero gravity.

Mass can be measured as inertial mass by placing masses in an oscillating system and correlating period information with mass information. The system would have to be calibrated with known masses before the mass of an unknown could be determined. One such an oscillating system is an Inertial balance, shown below.

One end of the balance is clamped to the table and a series of standard weights are taped to the other end. The free end is set in motion and the periods of the motions are determined from the time for a known number of oscillations. [Period is defined as the time for one complete oscillation, or total time for all oscillations divided by the total number of oscillations.] A hole is provided in the oscillating end to hold an unknown mass which can oscillate with the apparatus.

Apparatus

Besides the Inertial balance, you will also need:

Six 100 gram masses ( or equivalent)

1 C-clamp and unknown mass

A stopwatch or timing device

Masking tape

And finally; string, a ring stand support, three-way clamp, and a rod for the last step.

Procedure

1. Clamp the inertial balance to the table using the c-clamp, the clamp should be centered so that the free end of the balance can oscillate horizontally without hitting the clamp.

2. Tape a 100 gram mass to the free end of the balance and pull the free end about 4.0 cm away from equilibrium and release. The balance should oscillate smoothly after released. Note: if the original displacement is too large and the oscillations are inconsistent, the results of the experiment may not meet expectations.

Use the timing device to determine the total elapsed time for 50 complete oscillations of the balance. An oscillation must be a complete round trip, regardless of the starting point. Record the time in your lab notebook. Generally, because of reaction time error, it is not recommended to start timing when the balance is initially released.

3. Repeat the timing procedure until you have three time values for this mass. Later you can compute the average of the three values and use this number to determine the period.

4. Repeat the timing measurements for additional masses on the platform, increasing by 100 grams each time until 600 grams has been used.

5. Now, put the unknown mass in the hole and get three measurements for it as well.

6. Using the ring stand support and string, hang the unknown mass from the support so that its weight is supported by the string yet it is still sitting in the hole in the platform (the mass should be no more than 1.0 mm above the platform) Then get three time trials for this suspended mass and record as before.

7. Finally, get three time values for oscillations of the empty balance and record as before.

8. Now, use your ingenuity to find a way to oscillate the balance vertically instead of horizontally. Take one measurement of 50 oscillations with 300 grams taped to the end.

8. The last data value needed is the gravitational mass. Use one of the laboratory balances to obtain this value.

9. Compute all average times, periods and period-squared values (or let the spreadsheet do it for you). Be sure to show a sample calculation near your computed values.

Calculations

1. Using only the data for the known masses, construct a graph (see spreadsheet) of period-squared versus mass (mass on the x-axis). Mass should be in units of grams and period2 in units of s2. Determine the slope and y-intercept of the best straight line through the data and record these values. These two values are the m and b of equation [2]

Note: if you use the spreadsheet for calculations and graphs, you do not need to show how the slope and y-intercept of the graph are determined. However, any results calculated from these values should be included in sample calculations.

2. Plug in your calculated periods for the unknown (unsuspended and suspended) into equation [2] to find the inertial mass in each case. See your instructor for additional help if you need it.

3. Compute the percent errors in the calculated inertial masses using the gravitational mass as the accepted value. Show one sample and record in the appropriate location.

Questions

Advance Study (these relate to process and expected results)

1. What is inertia?

2. How is weight different from mass? How are they similar?

3. Why is it suggested to not begin timing at the initial release of the inertial balance?

4. Could this experiment be done in the “weightlessness” of outer space? Explain.

Post-Lab (these relate to results and analysis thereof)

1. Compare the suspended and unsuspended results for inertial mass. What do you see as the cause for the differences in these values?

2. In the data, the mass of the inertial balance itself was not included. If you could find a way to include it, how would it affect your calculated results for the inertial mass?

3. Equation [2] is only accurate for an object moving in one linear dimension. Did the inertial balance move in one dimension or two? Explain.

5. How did the vertical oscillation trial compare to the horizontal data. Would the orientation matter if the vibrations were done in space?

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