Answer all of the following questions. Do not send in a partially completed assignment. Answer each question as completely as possible, but do not merely copy answers from your reading materials. If you cite a reference in the text, explain the meaning of the citation.
(1) To calculate the acceleration due to gravity by observing the motion of a pendulum
(2) To investigate the effect of varying mass on the period of a pendulum
(3) To investigate the effect of varying the length of a pendulum on the period
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.
The period of the pendulum (T) is the time that takes for the pendulum to complete a whole cycle; that is, to swing along its path and return to the initial state. The calculation of this period is based on a complex equation. However, if the initial angle of displacement of the pendulum with respect the vertical is less than 30°, the equation can be simplified as:
with T being the period, L the length of the pendulum and g the acceleration due to gravity (9.81 m/s2 on the surface of the Earth). As the initial angle of displacement increases, the error between the period predicted by Equation (1) and the measured period also increases. The figure below shows the diagram of the basic variables of a pendulum:
In this diagram, each number represents the following concepts:
1 Bob with a mass: location of highest potential energy and lowest kinetic energy
2 Pendulum at equilibrium: location of highest kinetic energy and lowest potential energy
3 Bob’s trajectory
4 Angle θ
5 String or rod (in equations for this lab, this is assumed to be massless)
6 Pivot point (in equations for this lab, this is assumed to be frictionless)
7 Amplitude: distance between points 1 and 2
During the cyclic swinging motion of a pendulum, there is a constant yet gradual exchange between kinetic energy and potential energy. In order to describe this phenomenon, some terms should be defined:
Bob: The mass on the end of the pendulum.
Cycle: One swing of the bob back and forth.
Displacement: The distance from the pivot point straight down to the bottom of the bob. See the dotted line between #6 and #2 in Figure 1.
Period (T): The length of time the bob requires to swing back and forth.
Periodic motion: This is a motion in which the object returns to the point of origin repeatedly.
Frequency: The number of complete cycles per unit of time. In Figure 1, this is illustrated as the path the bob takes starting at position 1 and returning to position 1 over a period of time.
Amplitude: The distance the pendulum travels from the center point out to the point of maximum displacement. See #7 in Figure 1.
In this lab we will change three parameters of the pendulum apparatus and will evaluate how these changes affect its period.
Changing the amplitude of the oscillation
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. 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.
1. Attach a small plastic bag to the spring scale.
2. 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”.
3. 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.
4. 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 below as an example of the correct placement of the protractor.
5. 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.
6. Displace the bob out to the 5° 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 Table 1.
7. 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 Table 1 for Trial 1. We use 5 cycles in order to increase the accuracy of our data.
8. Repeat the experiment for Trials 2 and 3.
9. Calculate the Average and Standard Deviation for the period it takes the bob to complete 5 cycles.
10. Divide the Average value found by 5, in order to estimate the period for one cycle
11. Repeat this procedure, releasing the bobs at 10°, 15°, 20°, 25°, and 30°, and recording the results for each of the angles in Table 1.
|QUESTION 1 What initial angle yields the smaller standard deviation?|
|QUESTION 2Does the period of the pendulum’s oscillation change with the initial bob angle? Justify your answer.|
Using the approximated formula for the period of the oscillation, and the measures of period (T) found in Table 1, estimate the value of g for each initial angle. Assuming a value of g equal to 9.81 m/s 2, calculate the relative error for each initial angle. Record your results in Table 1.a.
|QUESTION 3For what initial angle do you obtain the lowest relative error? What about the highest relative error? What can you conclude from your data?|
Changing the mass of the pendulum
Remove washers from the pendulum’s bob until the mass of the bob is approximately 15 grams. We will repeat the same procedure as in Section 3.1, but to avoid excessive repetitions, we will only use an initial displacement of 10°.
Add washers to the to the pendulum’s bob until its mass is approximately 50 grams and repeat the previous measurements -only for an initial displacement of 10°.
Complete the data in Table 2. You can use the data you obtained in Table 1 for the mass of 25 g
|QUESTION 4What is the effect of the bob’s mass on the period of the pendulum’s amplitude? Did you expect these results? Explain.|
Changing the length of the string
For this section we will use a mass of the bob of approximately 25 grams. We will also use a single initial displacement angle of 10°.
We will use the procedure as before to measure the period of the oscillation. In this section, we will change the length of the string to 0.75 m, 0.50 m and 0.25 m. Complete Table 3 (once again, you can use the data from Table 1 to complete the row for 1.0 m).
|QUESTION 5What is the effect of changing the length of the string? Did you expect these results? Explain.|
Using the same equation as in Section 3.1, we will estimate again the value of g. Complete Table 3.a below.
|QUESTION 6What length of string yields the smaller relative error? Explain your results.|
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