Oscillations of an Obstructed Pendulum

Aim:

To investigate how – in an obstructed pendulum – varying the distance from the top of the pendulum to the obstruction affects the time period of one horizontal oscillation, and to use this information to verify the relationship shown below. After verifying the relationship, the value of ‘g’ must be derived from the graph.

The mass, physical shape and physical properties will be kept constant. The positioning of the stand, the orientation of the obstruction, and the angle of displacement will be kept the same. In addition to this, the mass and physical properties will be kept the same. The room conditions will be minimized, and external wind will be kept to a minimum.

The equation that must be verified is:

t=-π2gdt+2π lg

Where

g = gravitational acceleration (ms-1)

d = distance from top of pendulum to obstruction (m)

t = time taken for one oscillation (s)

Variables:

Independent Variable:

The distance from the top of the pendulum to the obstruction (40cm, 45cm, 50cm, 55cm, 60cm, 65cm, 70cm).

Dependent Variable:

The time period of one oscillation

Control Variables:         

• Mass and properties of the bob being used
• Angle of displacement (10°)
• Mass and physical properties of the string being used
• Positioning and location of the stand
• The orientation of the obstruction
• Room conditions

Apparatus:

• 1 stand to which the spring, the bob, and the obstruction will be attached
• 1 bob of mass 100g
• 1 string
• 1 wooden rod of approximately 15cm length (obstruction)
• 1 stopwatch
• 1 metre ruler

Risk Assessment:

1. The stand must be placed on a stable surface so as to prevent it from falling and injuring a person.

32.00

31.98

1.60

2

0.45

31.36

31.32

31.37

31.35

1.57

3

0.50

30.56

30.50

30.52

30.53

1.53

4

0.55

30.09

30.10

30.13

30.11

1.50

5

0.60

28.98

29.06

28.99

29.01

1.45

6

0.65

28.36

28.39

28.41

28.39

1.42

7

0.70

27.41

27.88

27.42

27.57

1.38

Calculating Uncertainties

To calculate the time period of 1 oscillation of the pendulum, one must first know the uncertainty of each measuring instrument used. The uncertainty of any instrument can be found through:

Uncertainty = Least Count2

This results in a least count of ±5.0 x 10-4 m for the meter ruler. The uncertainty of the stopwatch was calculated to be ±0.05seconds.

Now, to find the uncertainty of the time period of 1 oscillation, we must know that the time period = Average amount of time taken for 20 oscillation TAVG20

TAVG has an uncertainty of ±0.05 seconds.

Now, when we divide TAVG by 20, the following equation must be used to find the uncertainty:

Δ = uncertainty

x = TAVG

y = TAVG

R = value of TAVG ÷ 20 = Time period of one oscillation

ΔT = ±T (Δxx+ Δyy)

Calculations:

The relationship that we have to verify – as mentioned before – is as following.

t=-π2gdt+2π lg

As we must find a linear relationship, we need this equation to be made into the form y = mx +c. Therefore, let t = y and let dt= x. Now, the equation will look like the following:

y=-π2gx+2π lg

Now, it can be seen that this is a linear equation with gradient of -π2gand y-intercept of 2π lg

Now that we have this equation, we must plot dtin the x-axis and‘t’ in the y-axis.

The graph that follows on the next page shows the relationship between dt

An easier, and more practical approach may be to use a different, more advanced set of stopwatches. These stopwatches would have increased respond time, and would respond to human input (pressing the ‘start’/’stop’ button) in a better and faster way. This would in turn increase the precision of the results, and would help make the results more reliable.

In addition to this, instead of 3 trials being conducted for each length, 5 trials should have been conducted; this would have resulted in more tests of the theory, and ultimately would have had the effect of creating more reliable, accurate, and valid results.

Also, at times, the standing clamp that was used got out of position, and became positioned at an angle. This would be an easy error to eradicate: standing clamps which are firmer, more rigid, and are less likely to move and shake during the experiment should be used. This will mean that the position of the stand will remain constant, and this is very important to the accuracy of the results.

To conclude, there were a few random and systematic errors present during this experiment, and these could have had a large impact on the final results. If these errors were eliminated, the experiment would become more reliable, and therefore, the results would become more accurate. This would mean that errors which may render a calculated gravitational acceleration at 11.61ms-2 would be eliminated, thus creating a better and more practical experiment.