# Science

Student’s Name: Vickie Gray Date of Experiment: 21 Oct 2013

Date Report Submitted: 27 Oct 2013

Title: EXPERIMENT 11: Pendulum and the Calculation of g

Purpose:

The objective of this experiment is to let the students make an experimental calculation of acceleration to gravity (g) by observing and getting some parameters of pendulum.

Procedure:

A weight bag of mass 25.1 grams was connected to a fixed support via a 101 cm string attached to the bag that acted as a bob for the pendulum set-up. Protractor was placed behind the string for amplitude measurement. The meter was placed horizontally with the 50 cm mark behind the bob at rest. The bob was pulled out at different angles (5, 10,15,20,25 and 30) and the time for 5 cycles was measured for each of those angles. These trials were done 3 times for each angle. The average time and period were measured. On the next set-up, the weight bag was replaced with another with mass two times its weight (52.1 kg). Three trials were made at 10 degrees and 101 cm and the average time and period were computed. On the last set-up, the original 25.1 grams weight bob was used as the final bob. By pulling it out at 5 degrees, three trials were made for varying length at 0.25, 0.5, 0.75 and 1 meters and the average time and period were computed. The experimental values of g were obtained with g = 4L / T2 and the result was evaluated with the theoretical value of g equal to 9.81 m/s2.

Data Tables:

DATA TABLE 1

Length of string: 101 cm = 1.01 m

Mass of bob = 25.1 g = 0.0251 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 cycles

5

100.6

10.1

10.3

10.4

10.267

2.054

10

99.5

10.5

10.4

10.4

10.433

2.087

15

97.5

10.9

11

11.1

10.967

2.193

20

94.9

11.1

11.3

11.5

11.3

2.26

25

91.5

11.5

11.7

11.9

11.7

2.34

30

87.5

12

12.2

12.3

12.167

2.43

DATA TABLE 2

Length of string: 101 cm = 1.01 m

Amplitude: 10 deg

Bob weight

Trial 1

Trial 2

Trial 3

Avg. Time

Period

Grams

50.2

10.5

10.6

10.6

10.567

2.113

DATA TABLE 3

Mass of bob: 25.1 g

Amplitude: 5 deg

Length(m)

Trial 1-sec

Trial 2 -sec

Trial 3 -sec

Avg. Time

Period

0.25

5.1

5.2

5.1

5.133

1.027

0.5

7.2

7.3

7.4

7.3

1.46

0.75

8.6

8.5

8.7

8.6

1.72

1

10.1

10.3

10.4

10.267

2.053

Observations:

All measurements are recorded above on the Data Tables section. On Data Table 1, it can be observed that as the amplitude is increased, the period also increased by a negligible amount. On Data Table 2, it can be observed that doubling the mass at 50.2 grams has negligible effect on the period. On Data Table 3, increasing the length increases the period significantly.

With all of these having said, it is wise enough to say that angles at small values and mass did not affect the period. It was the length that affected the period.

The student would like to add the following computation for the experimental value of g at varying length.

For Data Table 3 (varying length)

T = 2√(L/g)

g = 4L / T2

at L = 0.250 m T = 1.027

g = 4L / T2

g = 4 (0.250) / (1.027)2

g = 9.357 m/s2

at L = 0.500 m T = 1.46

g = 4L / T2

g = 4 (0.500) / (1.46)2

g = 9.260 m/s2

at L = 0.750 m T = 1.720

g = 4L / T2

g = 4 (0.750) / (1.720)2

g = 10 m/s2

at L = 1 m T = 2.053

g = 4L / T2

g = 4 (1) / (2.053)2

g = 9.367 m/s2

We will just use the average of the g for 4 trials as a basis for our comparison with the theoretical value of g which is 9.810.

Average = (9.357 + 9.260 + 10.000 + 9.367) / 4

Average = 9.496 m/s2

% Error = | Experimental – Actual| /Actual

% Error = | 9.496 – 9.810 | / 9.810

% Error = 3.2 %

Questions:

A. How did the change in the weight of the bob affect the resulting period and frequency?

Adding the weight increases the period by a very small amount in the experiment. But this difference is negligible (0.0264 seconds) as theoretically mass should not affect the period in accordance with T = 2pi√ (L/g). Thus it is enough to say that the mass has no bearing with period.

B. How did the change in amplitude affect the resulting period and frequency?

It did not affect the resulting period and frequency that much. As the amplitude increased in degrees, the period also increased a little bit. Theoretically speaking, as angle increases the period obtained by using the equation L = 2pi√(L/g) is less reliable.

C. How did the change in length of the pendulum affect the period and frequency?

It significantly affected the period. As the length of the pendulum was increased, the period increased. .

D. What would happen if you used very large amplitudes? Check your hypothesis by

trial. What amplitude did you use? What is the result?

Hypothesis: The period increases since the angle is increases.

At 80 degrees, L =101 cm, and m=0.251 kg, the period is 2.9 seconds which is way higher than the theoretical value of T = 2*pi√(1.01/9.8) = 2.02 seconds

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?

Hypothesis: A magnet would make the travelling of the bob faster as it would have the same effect as increasing the gravitational force.

The result of the trial conformed to my hypothesis. The period decreased by a small amount which indicate faster travel time.

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?

The experimental and theoretical values of g were close enough. The error was 3.2 %. There are actually several reasons but probably the main contributor was air resistance. Equipment limitation and inaccuracy of measurements are amongst other reasons.

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?

At high altitude the acceleration due to gravity is decreased. In this case the gravitational force also is decreased. Having said these, we know that when these parameters are decreased, the period will increase as they are inversely proportional.

At weightless condition, the period becomes infinite as the force of gravity approaches zero. In this case the bob will remain as where it is raised.

Conclusions:

The acceleration due to gravity is 9.81 m/s2. This value can be approximated by observing and measuring the value of length of the string and the period of the pendulum according to the equation g = 4L / T2. Rearranging this equation, period is T = 2√ (L/g). Thus, period of the pendulum is not dependent on its mass and depends primarily on the length. However the approximation given by T = 2pi√ (L/g) is accurate only at small amplitude. Increasing the amplitude increases the inaccuracy of the calculated period. This independence of period with mass and small angles and the dependence to length were all verified in the experiment.