Investigating the period of a simple pendulum and measuring acceleration due to gravity.

gradient

So the gradient can be used to calculate the acceleration due to gravity.

THE APPARATUS:

* String.

* Bob.

* Stop clock.

* Wooden blocks.

* Clamp stand.

* Meter ruler.

To ensure more accurate reading for length and time, I use:

* A meter ruler with minimum reading of 1mm (resolution = 1mm).

* A stop clock of minimum reading 0.01 seconds (resolution = 0.01sec).

* A wooden clamp to reduce the effect of friction.

* I would also take the time period for 20 oscillations so that I can get an accurate reading for T.

METHOD:

* Set up the apparatus as shown above.

* Allow the pendulum to stand freely and when it in its equilibrium position put a mark on the bench directly beneath the pendulum bob. This mark can be used for timing.

* Measure the pendulum string using the meter rule so that it is 0.10m.

* Displace the pendulum through a small angle and release it.

* Measure the time taken for 20 complete oscillations i.e the time taken to pass the point on the bench 20 times.

* Repeat it once more.

* Repeat the above steps for different lengths of the pendulum.

* Find the mean time for each oscillation and square it to find T². Tabulate the results.

* Draw a graph with T² in the y-axis and L in the x-axis.

* The gradient of the graph can be used to calculate acceleration due to gravity.

PRECAUTIONS:

The measures I should take to obtain accurate results:

. Make sure the angle is small (1/6 of the average length of the pendulum).

2. Measuring the length.

Care should be taken when measuring the length. It should be measured from the centre of the bob to the fixed end.

3. Measuring the time period.

4. Take a total of 40 oscillations (20 the first time and 20 the second time) to ensure more accurate timing.

5. Make sure the string is fixed well in the other end.

RISK ASSESSMENT:

The Experiment I am going to do does not have too many risks and it can be carried out in classroom conditions. But in order not to hurt others and myself while carrying out this experiment a few safety steps will be followed.

* The Clamp stand will be placed away from the edge of the table.

* If the apparatus are placed to close to the edge of the table then there is a high chance of it falling off the table and onto somebody legs'. This could hurt that individuals leg and so as not to cause any physical harm the clamp should be placed away from the edge of the table.

* A sensible distance maintained from the swinging bob.

* When I count the number of oscillations the bob is swinging it is better that I keep a safe distance between my face and the swinging bob. It is better for me to keep a safe distance because if the swinging bob hits my face then it would be very painful.

* Make the sure the string is held in place securely.

* Before I start the experiment I will make sure that the string is firmly secured between the two wooden blocks. If the string is not held securely then it might detach from the wooden block and since the string has the bob attached to it, it might hurt anyone nearby.

When I carry out my experiment I will make sure that I follow all these safety precautions to provide a safe environment to work in.

IMPLEMENTING:

PROCEDURE:

Before I started the experiment I made sure that I had drawn an appropriate table to record my results.

I then made sure that I set my apparatus up properly like the one shown in the diagram in the method. While I was setting it up I made sure that the string was clamped tightly between the wooden blocks. If it was not tightly clamped then the length wouldn't be reliable.

After setting up the apparatus I made sure that the length of the pendulum was 0.10m. I also made sure that I measured the length from the centre of the bob to the fixed point. I then moved the pendulum through a small angle and released it. I started the stop clock when I released the pendulum. I noted the time it took to complete 20 oscillations. I took 20 oscillations as the time taken to pass the mark on the table twenty times in the same direction.

I found the time period for eight different lengths of the pendulum. For each length I found the time period for 20 complete oscillations. I found the time period twice for each length to increase the reliability of my results.

Length (m)

Time for 20 oscillations(s)

Time for 20 oscillations(s)

0.10

13.37

13.13

0.30

23.09

23.02

0.50

28.38

28.30

0.70

33.38

33.94

0.90

37.78

37.69

1.10

41.53

41.56

1.30

44.78

44.44

1.50

48.26

48.59

ANALYSIS:

RESULTS:

Length (m)

Time for 20 oscillations(s)

Time for 20 oscillations(s)

Mean time(s)

Time² (s²)

0.10

3.37

3.13

3.25

0.44

0.30

23.09

23.02

23.06

.33

0.50

28.38

28.30

28.34

2.01

0.70

33.38

33.94

33.66

2.83

0.90

37.78

37.69

37.74

3.56

.10

41.53

41.56

41.55

4.32

.30

44.78

44.44

44.61

4.98

.50

48.26

48.59

48.43

5.86

Gradient of the graph = ?y = ?T²

?x ?L

= 5.20

1.30

= 4 ms¯².

But

m = 4?²

g

g = 4?²

4

g = 9.87 ms¯².

ERRORS:

Percentage uncertainty = The uncertainty (resolution) × 100%

Average value

* Length:

Average value: (0.10 + 0.30 + 0.50 + 0.70 + 0.90 + 1.10 +1.30 + 1.50)

8.00

= 6.40

8.00

= 0.80m

= 800mm.

Resolution = 1mm

* Percentage uncertainty = 1 × 100%

800

= 0.125%.

* Time period:

Average value = (13.25 + 23.06 + 28.34 + 33.66 + 37.74 + 41.55 + 44.61 + 48.43)

8.00

= 270.64

8.00

= 33.83s

Resolution = 0.01s.

* Percentage uncertainty = 0.001 × 100%.

33.83

= 0.00296 × 2

= 0.00562%

* Percentage error = The value I got - The actual value × 100%

The actual value

The value I got = 9.87ms¯².

The actual value = 9.81ms¯².

* Percentage error = 9.87 - 9.81 × 100%

9.81

= 0.06 × 100%

9.81

= 0.612%.

CONCLUSION:

As expected I get a straight line graph through the origin. The time period which I got from carrying out this experiment seem quiet accurate since none of the points in the graph are too further away from the straight line.

The only points that are a bit further away from the straight line are the first three points. But since the period depends on the length, so for short lengths the period will be small and the error in our time period will be relatively large.

There are no anomalous results for me to comment on since all the points are close to the straight line. This maybe because the equipment I used to measure the length and the time period are quiet accurate. This is proved by the fact that when I worked out the percentage uncertainty of the length it was only 0.125% which is a very small amount. The percentage uncertainty for the time period worked out to be 0.00592%. Even this is a very small value. So this shows that to an extent the equipment I used are accurate enough to give reliable results.

The percentage error worked out to be 0.612%. Since this is a very small value it shows that the results I got are quiet accurate to give a reliable value for acceleration due to gravity.

EVALUATION:

The experiment I have done is quiet accurate in that the percentage error is only 0.612%. This shows that the results I obtained by carrying out this experiment are reliable.

The causes of anomalous results could be due to air resistance. The air resistance will gradually reduce the amplitude of the pendulum and this could lead to unreliable results. To overcome this problem in an ideal world I would perform my experiment in vacuum.

The other factor which could have lead to unreliable results is the friction between the string and the wooden blocks. The friction would increase the time period and thus this could lead to unreliable results.

The other factor which could have lead to me getting unreliable results could be the fact that I had to start the stop clock simultaneously when I released the pendulum. But I found hard to start the stop clock and release the pendulum at the same time as well.

I was only able to repeat my experiment twice due to shortage of time. To get even more accurate results I could repeat the experiment couple more times.

If there had been more time then I could have carried out further extension work on the experiment. I could have:

* Investigated the effect of the displacement angle on the period of the pendulum. I would however expect that for angles between 0°to 15° the results will be relatively accurate. Typical values are:

Displacement (m)

Displacement angle (°)

Time for 20 oscillation (s)

Mean time (s)

0.05

2° 50'

40.40

2.02

0.10

5° 40'

40.40

2.02

0.16

9° 0'

40.40

2.02

0.21

2° 0'

40.40

2.02

0.25

4° 20'

40.40

2.02

0.30

7° 20'

40.40

2.02

0.40

23° 20'

40.40

2.02

This shows that the period of a simple pendulum is independent of the angle of the swing (for small angles).

If there had been more time then I could carried out an experiment with different displacement angles and found the time period.

* Investigated the effect of the mass of the bob.

I could have found the time period for a simple pendulum for different masses of the bob for the same material.

However I don't think this would effect the time period of a simple pendulum because

T = 2? L

g

The equation doesn't have mass in it so changes in mass of the bob would not effect the time period of a simple pendulum.

All in all the experimental results that I got for the time period of a simple pendulum is reliable and the value of acceleration due to gravity works out to very close to the actual value. This shows that the experimental method and the apparatus I used are quiet accurate to give reliable sets of results.

SHALINI THARMABALAN.