The Simple Pendulum PHY102 Equipment: Universal Table Clamp Long Rod Pendulum Clamp Steel Pendulum Wood Pendulum Lead Plumb Pendulum Photogate Timer and Power Supply Meter Stick Tape Measure Protractor Objectives: • • • Use the scientific method to determine which physical properties control the period of a simple pendulum. Estimate the value of g, the acceleration of gravity. Verify the dependence of the period on the initial angle of the pendulum. Background: When set out of its vertical rest position by an angle θ and released, the pendulum oscillates back and forth. The time needed for a complete oscillation (back and forth) is the period (seconds). Pivot Initial angle – θ Length (L) Mass (M) Mg Figure 1 A Simple Pendulum An examination of Figure 1 suggests that the period could depend on the mass (M), the length (L – from the fixed point to the center of the mass), the initial angle (θ) and the value of gravity. Edited for PHY102 by Professor Barbara Medvar For small initial angles (less than about 10 degrees), one can show (see textbook) that the period T is given by T = 2π (1) T2 = (4π2/g) L (2) Raising (1) to the square, one gets i.e. T2 should be a linear function of L and the slope of the line is equal to (4π2/g) The value of g can thus be estimated as g = (4π2/slope) T = 2π ( [1 + sin2 ( ) + . . . . (3) ) (4) where the infinite series in θ has been limited to its first term. Following the scientific method approach, we will conduct a series of experiments in which one parameter (mass, angle or length) is varied at a time while the others are maintained constant. Obviously, g cannot be varied. Setup: 1. Attach the Universal Table Clamp to the table, tightening the screw only until snug. Insert the long rod into the clamp and secure. 2. Attach the pendulum clamp to the long rod. Make sure the thumb screw is tightened onto the rod, and not alongside it. Edited for PHY102 by Professor Barbara Medvar 3. Attach the wood pendulum to the pendulum clamp, running the string under the clip. Make sure the string is about 14-18 inches long, and above the level of the screw holding the clamp to the table. 4. Tighten the screw holding the string. 5. Set up the photogate timer so that the pendulum will break the beam on the photogate. Plug in the power supply and set the timer to millisecond (ms) and [pend]. Edited for PHY102 by Professor Barbara Medvar Procedure: 1. Influence of mass: a. Choose one length (L), say 40 cm, and hang a wooden ball. b. Using a protractor, set the pendulum 10 degrees from the vertical and release. c. Record the time (period). Repeat twice and report the average on the data sheet. d. Repeat a. to c. above for a steel ball, using the same 40 cm length. 2. Influence of angle: a. Use the steel ball and a length of 40 cm (same as above) b. Set the angle of release at 5 degrees from the vertical and release. c. Record the time (period). Repeat twice and report the average on the data sheet. d. Repeat a. to c. above for angles of 10 (already done in 3.1), 15, 20, 25, and 30 degrees from vertical. 3. Influence of length: a. Use the steel ball and an angle of 10 degrees in all cases. b. For lengths of 20 cm, release the ball from 10 degrees from the vertical. c. Record the time (period). Repeat twice and report the average on the data sheet. d. Repeat a. to c. above for the lengths of 40 cm (already done), 60 cm, 80 cm, 100 cm, and 120 cm. e. Draw a graph of period squared versus length. Perform a regression analysis to determine the slope of the best fit line and its correlation coefficient. f. Estimate the value of g using equation (3). g. Calculate the error between your experimental value of g and the standard. Discussion/Conclusion Discuss the various aspects of the experiments and whether the objectives have been achieved. Use the data generated to support your discussion and use full sentences. Edited for PHY102 by Professor Barbara Medvar The Simple Pendulum – Data Sheet PHY102 Name: Date: Influence of the mass: Length: 0.4 m Angle: 10 degrees Mass Period: T (s) Wood sphere 0.9203 Steel sphere 0.595 Influence of the angles: Mass: Steel Length: 0.4 m Angles (degrees) Period: T (s) 5 1.0156 10 0.6807 15 0.6546 20 0.6888 25 0.6768 30 0.6617 Influence of the Length: Mass: Steel Angle: 10 degrees Mass (cm) Period: T (s) 20 0.4696 40 0.6989 60 0.7829 80 0.8519 100 1.0105 120 ----------- Edited for PHY102 by Professor Barbara Medvar The Simple Pendulum – Data Sheet PHY102 Name: Date: Period versus Initial Angle: Angle (degrees) 5 Angle (radians) 1 + (¼) sin2 (θ/2) 10 15 20 25 30 Edited for PHY102 by Professor Barbara Medvar T (s) T/T5 (T/T5)/ [1 + (¼) sin2 (θ/2)
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