EXPERIMENT 11: Pendulum and the Calculation of g Read the entire experiment and organize time, materials, and work space before beginning. Remember to review the safety sections and wear goggles when appropriate. Objective: To calculate the acceleration due to gravity by observing the motion of a pendulum. Materials: Student Provides: Support for the pendulum Weights, coins, or washers Small plastic bags Tape From LabPaq: Meter tape Stopwatch Protractor String Spring scale Discussion and Review: A pendulum is a weight hanging from or supported at a fixed point so that it swings freely under the combined forces of gravity and momentum. A typical simple pendulum consists of a heavy pendulum bob (mass = ) suspended from a light string. It is generally assumed that the mass of the string is negligible. If the bob is pulled away from the vertical with some angle, , and released so that the pendulum swings within a vertical plane the period of the pendulum is given as: Equation 1: where gLh is the length of the pendulum and ggh is the acceleration due to earth’s gravity. Note that only the first three terms in the infinite series are given in Equation 1. The period is defined as the time required for the pendulum to complete one oscillation. That is, if the pendulum is released at some point, P the period is defined as the time required for the pendulum to swing along its path and return to point, P. The above formula for the period of the pendulum is greatly simplified if we limit the initial angle ƒÆ to small values. If ƒÆ is small we can approximate the period of the pendulum with a gfirst-order expressionh, which in the case of our simple pendulum is given as: Equation 2: Hands-On Labs SM-1 Lab Manual 88 Note that the period in this expression is independent of the pendulum’s mass at initial angle, ƒÆ. Also, it is important to understand that the above equation is valid only for small angles and is substantially less accurate with large angles. During the cyclic swinging motion of a pendulum there is a constant yet gradual change of kinetic energy to potential energy and back to potential kinetic energy. In order to describe this phenomenon here are some terms you need to know: Amplitude: The distance the pendulum travels from the center point out to the point of maximum displacement. Frequency: The number of complete cycles per unit of time. Periodic motion: The type of motion in which the object returns to the point of origin repeatedly. Because of the rotation of the earth a pendulum will be slightly deflected on its course on every circuit. This is observable on a very long pendulum called a Faucault pendulum. Look up Faucault pendulum. Period (T): The length of time for one trip, back and forth. Displacement: The distance from the center point, straight down. Cycle: One swing of the pendulum back and forth. Bob: The mass on the end of the pendulum. PROCEDURES: Before beginning, you must first find a suitable support from which to freely hang your pendulum. Ideally there should be a wall close behind the support so you can easily affix your protractor and meter tape for recording movements. A bathroom or kitchen towel bar is ideal for this purpose. Or you might rig a support like the one at right and place it on a narrow shelf or table top. The important things are that your support allows the pendulum to hang freely; that you are able to read and record measurements from the protractor and meter tape; and that pendulum string not touch anything or be obstructed from any direction. You will also need to make a weight bag to use as the bob. Place coins, weights, or washers totaling around 25 grams inside a small plastic bag. Tie a short piece of string around the top of the bag so the weights cannot fall out Weigh your bob and record the weight. Note: one quarter should weigh around 5.7 grams. 1. Weigh your bob and record its mass. Hands-On Labs SM-1 Lab Manual 89 2. Suspend the bob from a string that measures exactly one meter (100 cm) between where it attaches to the support and where it attaches to the center of the weight bag you are using as a bob. To accomplish this, you obviously must start with string that is longer than a meter. 3. Securely affix a protractor behind where the string is attached to its support so you will be able to measure the pendulumfs amplitude in degrees. 4. Stretch a meter tape horizontally and securely affix it so that its 50-cm mark is directly behind the bob at rest. 5. Observe the protractor and pull the bob out to the 5o-mark. Then observe the meter tape and record the distance in cm of the bob displacement. 6. With a stopwatch in your other hand, release the bob and time how long it takes for the bob to move through 5 complete cycles. Record the time in Table 1. Perform two more trials from the 5o-mark. Record each time, then average the three trials and calculate the period for one cycle. 7. Repeat the procedure and record results for each of the angles shown in Table 1. DATA TABLE 1: Length of string: _____ cm = _____ m Mass of bob: _____ g = _____kg Amplitude Amp. Trial 1 – seconds Trial 2 – seconds Trial 3 – seconds Avg. Time Period Degrees cm 5 cycles 5 cycles 5 cycles 5 cycles 1 cycle 5 o 10 o 15 o 20 o 25 o 30 o 8. Place double the bob weight into a second plastic bag and repeat this procedure using a 10o.amplitude Record the data in Table 2. DATA TABLE 2: Length of string: ________ cm = _______ m Amplitude: _______o Bob Weight Trial 1 Trial 2 Trial 3 Avg. Time Period Grams 9. Put the original bob back on your pendulum. Use a 5o or 10o amplitude and make three trials each with successively shorter lengths of string, i.e., 100 cm, 75 cm, 50 cm and 25 cm. Record this data in Table 3. Hands-On Labs SM-1 Lab Manual 90 DATA TABLE 3: Mass of bob: ________ g = _______ kg Amplitude: _______o Length (m) Trial 1 Trial 2 Trial 3 Avg. Time Period .25 .50 .75 1.0 Calculations: Solve the pendulum formula for g. Substitute the data you recorded for the values for t and L (length of string) in the formula. Calculate to the correct significant figures. Then calculate your percentage error as compared to the accepted value for g. The accepted value of g is 9.8 m/s2. t = 2 ƒÎ ã(length/g) g = (2ƒÎ)2 L t2 where: g = acceleration due to gravity t = time in seconds L = length of pendulum string in meters Note: If you get very large errors in this lab you are doing something wrong. Your calculations need to be double-checked. Questions: A. How did the change in the weight of the bob affect the resulting period and frequency? B. How did the change in amplitude affect the resulting period and frequency? C. How did the change in length of the pendulum affect the period and frequency? D. What would happen if you used very large amplitudes? Check your hypothesis by trial. What amplitude did you use? What is the result? E. Hypothesize about how a magnet placed directly under the center point would affect an iron bob? Try it and find out. Did your trial verify your hypothesis? F. How close was your calculation of the value of g at your location? What might be a few sources for error in your experimental data and calculations? G. What would you expect of a pendulum at a high altitude, for example on a high mountain top? What would your pendulum do under weightless conditions?