Objective and Background

Objective:

The Objective of this experiment is to examine the simple harmonic motion and to determine the value of the acceleration due to gravity from the analysis of the period of the simple pendulum. [1] Background:

There are three equations that will be used to calculate the period of motion of the simple pendulum. They are the slope of the line of the graph of T² against L, and the gravity of the pendulum motion. The period of the motion is the time needed for one complete cycle that a pendulum bob swing from the initial position to the other end, and then back to the initial position. [1] The equation to calculate period is,

T = 2πLg

Where,

T = Period of the motion, measured in s.

L = Length of the pendulum, measured in cm.

g = Acceleration due to gravity, measured in m/s2.

The slope of the line in the graph of T² against L can be used to determine the gravity of the pendulum motion. It is because,

y = mx

m = T² L= 4π²g

m = Slope of the line in the graph T²/L.

Therefore, to find the gravity of the pendulum motion, we can use the slope of the graph.

The slope of the graph is given by the formula,

g = 4π²m

g = Acceleration due to gravity, measured in m/s².

Procedure and Observations

Materials:

* String

* Metre Stick

* Stop watch

* Stand

* Pendulum bob

Procedure:

1) Materials listed above were taken for the experiment.

2) The pendulum bob was tied tightly with the string.

3) The string with the pendulum bob was hung on the stand.

4) A meter stick was used to measure the distance between the centre of mass of the bob and the top of the string.

5) The distance was recorded in the observation table.

6) The pendulum ball was held at a distance from the centre and it was released.

8) A stop watch was used to time the time needed to complete ten cycles.

9) The time was recorded in the observation table.

10) Steps 4-7 were repeated four more times with different lengths.

Observations:

L (m)| 10T (s)|

0.185| 9.01|

0.39| 14.13|

0.595| 15.04|

0.79| 17.58|

1| 19.67|

Diagram of the Pendulum

Figure [ 1 ]

Calculations and Results

Method 1 – Graph of T2 vs. L

Data collected

L(m)| T2(s2)|

0.18| 0.81|

0.39| 1.99|

0.59| 2.37|

0.79| 3.09|

1| 3.86|

Hand drawn graph

∆x

∆y

Figure [ 2 ]

The slope can be determined by m=∆x∆y.

So, by taking a value for x

x = 0.4 cm

y must then be

y = 1.4 cm

m= 1.4 cm0.4 cm

m=3.5

The error would be given by

∆mm= ∆x1x12+ ∆x2x22

∆m= m 0.051.42+ 0.050.42

∆m= 3.5 0.051.42+ 0.050.42

∆m=0.45

The acceleration due to gravity is given by

g=4π2m

g=4π23.5

g=4π23.5

g=11.3 m/s2

Calculating the error for g would yield

∆gg= ∆mm2

∆g= g 0.453.52

∆g= 11.30.453.52

∆g= 1.45 m/s2

g=11.3 m/s2 ± 1.45 m/s2

Solving for the percentage deviation would give

% deviation= Actual value-Expected valueExpected value* 100%

Expected value=9.8 m/s2

% deviation= 11.3 m/s2-9.8 m/s29.8 m/s2*100%

% deviation= 11.3 m/s2-9.8 m/s29.8 m/s2*100%

% deviation= 15.3%

Method 2 – Linear Regression

Excel graph

Figure [ 3 ]

The equation of the line is given by T2 = 3.53L + 0.33

Where

m=3.53

The acceleration due to gravity is given by

g=4π2m

g=4π23.53

g=4π23.53

g=11.1 m/s2

Solving for the percentage deviation would give

% deviation= Actual value-Expected valueExpected value* 100%

Expected value=9.8 m/s2

% deviation= 11.1 m/s2-9.8 m/s29.8 m/s2*100%

% deviation= 11.1 m/s2-9.8 m/s29.8 m/s2*100%

% deviation= 13.2%

Conclusion

By comparing these two methods of calculating the acceleration due to gravity it is clearly noticeable that there is a difference between the two, when it comes to the accuracy. When calculating g using the hand drawn graph method it yielded =11.3 m/s2 ± 1.45 m/s2. However, when using the linear...