Mass Pendulum

Attributes linked to this design project (CEAB's Graduate Attributes):

> Knowledge base for engineering

> Problem analysis

> Investigation

> Design

> Use of engineering tools

> Life-long learning

METHOD:

Experiments I, II, and IIIb were conducted using a 35cm ± 0.1cm, 6.44g metal string, used to hang a 200 ± 0.01g mass. A metal wire was used throughout this project to provide greater strength when the load was attached.

For Experiment I, the data for the period was obtained by releasing the pendulum at an angle of about 30°, for Experiment II, the pendulum was released at about 80°, and for Experiment III, the pendulum was released at an angle of 50°. The pendulum's position was then obtained frame by frame using a tracking software “Tracker”.

A ruler was attached to the pendulum's structure to accurately calibrate the dimensions of the structure using the "Track" program and make calculations more accurate; likewise, a protractor was also attached to calibrate 0° in the software.

A clamp was used to hold the changed length of the metal string in place. A drawing pin could also be used to hold the string in place after changing the length of the string.

For Experiment IIIb, five different masses were used to test the effect of mass on the period of the oscillations. These five masses were: 100g, 150g, 200g, 250g, and 500g. Using masses with such values made more accessible and accurate calculations and readable graphs. Still, not all the masses had a pattern in terms of mass, allowing testing at far ends possible. To stay safe, weights with mass greater than 500g were not used.

RESULTS:

SECTION I:

Q FACTOR:

In this experiment, we were to find the pendulum's Q (quality) factor. If Q was much larger than 1, it measured how many complete oscillations it took for the system to decrease its amplitude to about 4% of its original amplitude. The oscillations were done from left to right, i.e., in the direction of the photograph.

Method 1 (Using equation):

One way to measure the Q factor is to measure the period (T) and the time constant of the decay (τ). The Q factor can then be calculated using the following equation:

Q=π(τ)/T

The values of tau and T were determined using a provided Python script after inputting values from the "Tracker" program:

τ: 45.47254948801433 +/- 1.8349536255307213

T: 1.2569781318924917 +/- 0.000222584579119616

The Q factor was calculated using the equation above and then adding the uncertainties (which were also calculated using the provided Python script):

Q = π (45.47) / (1.26) = 113.37 ± 4.56

Method 2 (Counting oscillations):

Another way was to count the number of oscillations until the amplitude is (e −π) ∼ 4% of the initial amplitude, and that value is the Q factor.

Alternatively, count the number of oscillations until the amplitude is e−π/2 ∼ 20%, and that is Q/2. When counted, the total number of oscillations was 55. The calculations were then made as follows:

Count = 55

Q/2=55→Q=110

3.1.3.1.1.3 Mathematical model:

where t is time, φ0 is the phase constant and τ and T are constants.

θ(t) = θ0 e −t / (45.473) cos (2π t/(1.257) + 0)

Here, φ0 is zero as the start time in the graph/calculation was the exact instant the pendulum was released.

Now, to calculate the value of T using the predicted model equation to calculate the period of the pendulum:

T = 2√0.35 = 1.18

3.1.1.2 SECTION II:

3.1.1.2.1 PERIOD VERSUS AMPLITUDE

This experiment investigated the effects of amplitudes on the period of the pendulum by measuring the changes in the period of the pendulum as done for Experiment I, the "Tracker" program was used to obtain the required data (as previously mentioned in the "Method" section of this report).

Now, after obtaining the values of the required parameters, we'll have a look at the power series:

where T0: 1.3440408751071489 +/- 0.461795831730099

B: -0.03626694870217612 +/- 0.06351308960499027

C: 0.09511702480884357 +/- 0.3948157824690088 T = (1.344) + (-0.036)Θ0 + (0.095)Θ02

As the value of B is smaller than its uncertainty, its value can be assumed to be 'experimentally zero.'

As seen in Fig. 3.1.11, a test for asymmetry was done by arbitrarily choosing one side as positive and releasing the pendulum from both the positive and the negative initial positions.

3.1.1.3 SECTION III:

3.1.1.3.1 PERIOD VERSUS STRING LENGTH

A power-law function was used to test the theory for this experiment experimentally:

Where k is theoretically equal to 2 and n is theoretically equal to 0.5 within the uncertainties. L0 would be zero if the effective length of the pendulum (from the pivot to the center of mass) were correctly determined.

After collecting data from ten different lengths and inputting those into a Python script, the following results were obtained:

Results of fitting data to f(x)=a*(b+x)**c:

a: 1.9993424433064624 +/- 0.0010808325869416959 (theoretically 2)

b: -0.0002448497804050666 +/- 0.0007008542546790206 (theoretically 0)

c: 0.49909499643776084 +/- 0.0014384337284851197 (theoretically 0.5)

where a = k, b = Lo, and c = n in the power law function.

3.1.1.3.2 PERIOD VS MASS

In Experiment I from Section I, the period (T) was theoretically calculated using the formula,

T=2√L

In Experiment I, L was considered the length of the spring, but in Experiment III, it is considered the distance from the pivot to the center of mass of the pendulum.

From Experiment I,

T = 2√0.35 = 1.18

From Experiment III,

T2 = 2√0.39 = 1.249,

where 0.39 ± 0.01m is the distance from the pivot to the pendulum's center of mass. 

T2 was considered the baseline as in Experiment I; the length of the string (at that time considered L) was 0.35m. After considering the center of mass, the length L was taken as 0.39m.

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