1.4.1 Defining Gravitational and Inertial Mass The mass of an object is defined as: a measure of the amount of matter it contains. There are two different quantities called mass: 1.4.1a Defining Inertial Mass Inertia is defined as: the property of all matter that causes it to resist being put into motion when a force is applied. Inertial mass is a measure of an object's inertia. An object with a small inertial mass begins moving more readily, and an object with a large inertial mass does so less readily. 1.4.1b Defining Gravitational Mass Gravitational mass is defined as: a measure of the strength of an object's interaction with the gravitational field. Within the same gravitational field, an object with a smaller gravitational mass experiences a smaller force (ie weight) than an object with a larger gravitational mass. 1.4.2 Inertial Mass and Newton s Laws of Motion To understand what the inertial mass of a body is, we consider Newton's Laws of Motion. According to Newton's second law, we say that a body has an inertial mass m if, at any instant of time, it obeys the equation of motion (1.8) where is the net (total) force acting on the body and is its acceleration. This equation illustrates how the inertial mass of an object relates to it s inertia. Consider two objects with different inertial masses. If we apply an identical net force to each, the object with a greater inertial mass will experience a smaller acceleration, and the object with a smaller inertial mass will experience a greater acceleration. In effect, the greater mass exerts more "resistance" to changing its state of motion in response to the net force. Page 1 of 10
1.4.3 Gravitational Mass and Newton s Laws of Motion The concept of gravitational mass rests on Newton's universal law of gravitation. Let us suppose we have two objects a and b, separated by a distance. The law of gravitation states: if two objects, a and b have gravitational masses and, then each object exerts a gravitational force on the other, of magnitude: (1.11) where G is the universal gravitational constant. This can also be stated as: if an object has a gravitational mass m at a given location in a gravitational field with field strength g, then the gravitational force on the object has magnitude: This is principle behind a gravitational balance. 1.4.4 The Equivalence of Inertial and Gravitational Mass The Galilean equivalence principle (a.k.a. the weak equivalence principle) is stated as: the inertial and gravitational mass of an object are equal. The first experiments demonstrating the equivalence of inertial and gravitational mass were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is unlikely to be true; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Roland Eötvös, using the torsion balance pendulum, in 1889. To date, no deviation from universality, and thus from Galilean equivalence, has ever been found. More precise experimental efforts are still being carried out. Page 2 of 10 Roland Eötvös 1848 1919
1.4.5 Using a Gravitational Balance On Earth, gravitational mass can be measured easily by using a beam balance. A beam balance uses the weight of an unknown mass to measure its gravitational mass. When you use a beam balance, you are comparing the force due to gravity on the unknown mass to that on the known masses. When the forces on each are the same, the balance remains horizontal and at this point, the weight of the unknown (and hence its gravitational mass) is the same as the known mass. 1.4.6 Using an Inertial Balance There are some situations where a beam balance cannot be used to measure the mass of an object. When in free-fall, for example in an orbiting Space Shuttle a gravitational balance won't work. Eg. #1. Why won t a gravitational balance work in orbit? Because the normal force is zero In situations such as these, an inertial balance must be used. An inertial balance uses the inertia of the unknown mass to measure its inertial mass. The unknown mass is anchored to the tray, then the tray is then set into oscillatory motion. The period of oscillation is recorded. An object with a greater inertial mass will oscillate with a longer period than an object with a lesser inertial mass. The result obtained is located on a graph of frequency vs. inertial mass for this inertial balance. In this way, the inertial mass of an unknown can be determined by interpolation. You can see an inertial balance at the museum of science and technology in Ottawa. This device is used by NASA to measure the mass of astronauts during spaceflight. Page 3 of 10
Notes regarding the proper completion of lab reports: 1. When writing up the lab, you must record answers to the analysis questions in full sentences. 2. When references have been used, they must be properly cited at the bottom of the page they are quoted on. Page 4 of 10
Name: Partner: /10 K/U /33 T/I /42 C /56 A Purpose: Part 1. How is an inertial balance calibrated? Part 2. How is a calibrated inertial balance used to find the inertial mass of an object? Apparatus: C-clamp inertial balance set of brass masses mass anchor stop watch test mass triple beam balance Procedure: Part 1: Calibration of an Inertial Balance 1. Anchor the ballast bolt to the balance pan, placing one washer below the pan and one above the pan. 2. Move the balance pan to the side and release it. 3. Measure and record the time for 10 complete oscillations making sure that the ballast bolt is not moving in the pan (The pan will be moving quickly so try to keep up!). 4. Repeat step 3 twice more for a total of three trials. 5. Anchor the first ballast mass into the pan using the ballast bolt. Repeat steps 2 4. 6. Repeat step 5 with successively higher ballast masses. Part 2: Determining the Inertial Mass of a Test Mass 1. Measure and record the time for 10 complete oscillations of the test mass making sure that it is not moving in the pan (Also make sure that the ballast bolt is present. 2. Measure and record the gravitational mass of the test mass using the gravitational balance. Page 5 of 10
Observations: Table #1.1 Mass and Time for Ballast Masses Mass (g) Trial 1 Time (s.) Trial 2 Time (s.) Trial 3 Time (s.) Table #1.2 Time for Unknown Mass Trial# Time (s) 1 2 3 Gravitational Mass of Unknown Mass: K/U T/I /10 C A Page 6 of 10
Analysis: 1. Complete the table of the mass (in grams) vs the average period (in seconds) for the inertial balance using the calibration data from table #1.1. Be sure to include the uncertainty of the average in your calculations. Table #1.3 Mass and Average Time for Ballast Masses Mass (g) Average Time (s.) Sample calculation of average time with uncertainty /30 A /10 KU 2. Graph each point in table #1.3 to form a calibration graph for the inertial balance you used. Make sure to include error bars. /10 K/U T/I /8 C /30 A Page 7 of 10
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3. Using the graph, determine the inertial mass of the unknown. /5 T/I 4. Using the gravitational mass as the expected value, calculate the percentage error in the mass of the test mass. Discussion: 1. Discuss three sources of experimental error and how they could be minimized. /6 A /9 T/I 2. Identify the controlled variables used during the lab. Remember, these are variables that you have some control over. 3. Identify the control trial(s) for this lab. /6 T/I /2 T/I K/U /22 T/I /10 C /6 A Page 9 of 10
Synthesis: 1. Explain why inertial balance is a good name for the device used in today s lab. /4 T/I 2. Identify two things that you have learned about inertial balances while completing this experiment. Extension: /4 T/I Consider the following scenario: The weight of the mass being massed by the inertial balance is supported by a string tied to a support with the mass still able to oscillate back and forth. Predict the effect this would have on the inertial mass recorded by the inertial balance and explain your prediction. Conclusion: /3 T/I Part 1. Explain the process of calibrating an inertial balance. Part 2. Explain the process of determining the mass of an object using a calibrated inertial balance /4 C /4 C K/U /11 T/I /14 C A Page 10 of 10