Discussion and review
A pendulum is a weight hanging from a fixed point so that it swings freely under the combined forces of gravity and momentum. A simple pendulum consists of a heavy pendulum bob (of mass M) suspended from a light string. It is generally assumed that the mass of the string is negligible. If the bob moves away from the vertical to some angle θ, and is released so that the pendulum swings within a vertical plane, the period of the pendulum is given as: T = 2π
Table 1: contents of Formula
|T||Period of a pendulum to complete one cycle|
|L||Length of string|
|g||Acceleration due to gravity: 9.81 m/s2|
Part 1: changing the amplitude
Before beginning, find a solid support from which to hang the pendulum. Ideally, there should be a wall close to the support so the protractor and tape measure can be attached for recording the pendulum’s movements. A bathroom or kitchen towel bar is ideal for this purpose.
A support similar to that shown in Figure 3 can be constructed and placed on a narrow shelf or tabletop. It is important not only that the support allows the pendulum to hang freely, but also that you are able to read and record measurements from the protractor and tape measure. Do not allow the pendulum string to touch anything or be obstructed from any direction. The pendulum apparatus must also be sturdy enough so that it does not bend, flex, or move in any manner as this will introduce error into the experiment. See Figure 4 for an example setup with the pendulum bob hanging from an over-the-door hanger.
- Attach a small plastic bag to the spring scale.
- Add washers to the plastic bag until the scale measures approximately 25 g total. The filled bag will hereafter be referred to as the bob. Record this value as “Mass of bob” in the place provided in Data Table 1.
- Measure a piece of string that is approximately 120 cm in length. Tie the string around the top of the bag so that the washers cannot fall out. Suspend the bob from this string so that it measures exactly 1 m (100 cm) between where it attaches to the support and the bottom of the bob.
- Use tape to affix the protractor behind where the string is attached to the support so you can measure the pendulum’s amplitude in degrees. The center hole in the protractor should be located directly behind the pivot point. The string should hang straight down so that the string lines up with the 90o mark on the protractor. See Figure 4 as an example of the correct placement of the protractor.
- Stretch the measuring tape horizontally and use tape to affix it to the wall or door so that its 50-cm mark is directly behind the bob at rest.
- Displace the bob out to the 5o mark and hold it there. Then observe the bob’s location during its first cycle as it swings relative to the tape measure and record the distance in centimeters as “Amplitude (bob horizontal displacement)” in Data Table 1.
- With a stopwatch ready to begin timing, release (do not push) the bob and begin timing how long it takes the bob to move through five complete cycles. Record this first trial time in Data Table 1 for Trial 1. Repeat the procedure for the second and third trials. Then average the three trial times to calculate the average period for one cycle, and record this value in Data Table 1.
- Repeat this procedure, releasing the bobs at 10°, 15°, 20°, 25°, and 30°, and recording the results for each of the angles in Data Table 1.
Length of string: _____ cm = _____ m Mass of bob: _____ g = _____kg
Data Table 1: Trial values at varying degrees
|Placement of BobDegrees||Amplitude (bob horizontal displacement) cm||Trial 1 (s) 5 cycles||Trial 2 (s) 5 cycles||Trial 3 (s)5 cycles||Avg. Time (s)5 cycles||Period 1 cycle|
IMPORTANT: The pendulum must swing without obstruction and should not strike the background as it swings.
Part 2: changing the mass
- Add more weights to the bag until the mass has doubled to approximately 50 g. Record this value as “mass of bob” in grams into the line provided next to Data Table 2.
- Repeat the procedure used in Part 1 using only a 10o amplitude for the starting point of the Record the data in Data Table 2.
Length of string: ________ cm = _______ m Amplitude: 10°
Data Table 2: Trial values for bob masses
|Bob weight (g)||Bob weight (kg)||Trial 1 (s)||Trial 2 (s)||Trial 3 (s)||Avg Time (s)||Period|
Part 3: changing the length of string
- Remove the weights until the original mass used in Part 1 (approximately 25 g) is inside the bag. Record this “mass of bob” in grams into the line provided next to Data Table 3.
- Put the original bob containing the washers back onto the pendulum. Use a 10o amplitude and perform three trials each with successively shorter lengths of string. For example, 1 m, 0.75 m, etc. Record the time in seconds into the columns labeled “Trial #1, 2, or 3 s” in Data Table
Mass of bob: ________ g = _______ kg Amplitude: 10o
Data Table 3: Trial values for string length
|Length (m)||Trial 1 (s)||Trial 2 (s)||Trial 3 (s)||Avg Time (s)||Period|
Part 4: Calculations
- Solve the pendulum formula for g using the values derived from this experiment. Equation 3 will be used in calculating “g.” Substitute the average data for time and the length of the pendulum into the formula. Calculate to three significant figures. Then calculate your percentage error as compared to the accepted value for g, which is 9.81 m/s2. See Equation 4.
- g = acceleration due to gravity
- t = time in seconds
- L = length of pendulum string in meters
Note: If you get very large errors, such as 20% or more, in this lab, double-check your calculations.
% error = experimental value – theoretical value × 100