What is the relation between time period length of a pendulum and acceleration due to gravity?

The pendulum motion is caused by a restoring force whose whose tangential component is opposite to displacement in direction. The tension from the string, if any, would always be perpendicular to the path. When there is no such force to provide the restoring force, the type of oscillation you mentioned would not happen.

BTW, a kind reminder- Mathematics is powerful friend of Physics, and a reliable ally also, in most of the cases. (Just a metaphorical illustration LOL) To clearly understand a problem, perhaps it's best to start at the physics, instead of the mathematics, since the equations were derived for particular scenario (e.g. in the presence of certain fields), which might be fundamentally different from the scenario at hand.

Extracts from this document...

Pendulum Laboratory Report

Investigation of the relationship between period and length for a simple pendulum and the determination of g

A simple pendulum consists of a point mass 'm', suspended from a fixed point using a mass less ideal string of length 'l', such that it can move forth and back from its mean position. When the simple pendulum is set in motion, it moves back and forth periodically. One complete to and fro movement of a pendulum about its mean position is known as an oscillation or vibration. The time taken for one oscillation is known as the time period (T).The time it takes to make complete oscillation is called the frequency 'ƒ' of the oscillation is the number of oscillation simple pendulum made in one second:  ƒ = 1 / T. Here are the laws of a simple pendulum:

• The period of a simple pendulum of constant length is independent of its mass, size, shape or material.
• The period of a simple pendulum is independent of the amplitude of oscillation, provided it is small.
• The period of a simple pendulum is directly proportional to the square root of length of the pendulum.
• The period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity. [1]

*where T = time for 1 cycle (s), l = length of pendulum (m), g = acceleration due to gravity (m/s2)

This study invests the relationship between the period and the length of a simple pendulum to determine the approximate value of the acceleration due to gravity g on planet Earth and planet X.

The virtual experiment was done online on 25th Febuary, 2012, at the Pendulum lab 2.03 (PhET, 2011) [2]. The apparatus was placed as shown in Figure 1. A digital clock at the toolbox was used to measure the time t for 10 complete oscillations. The pendulum bob was 1kg heavy and its mass was fixed throughout the experiment. Five different values of the string’s length were chosen to set the pendulum swing, which respectively were 0.5, 0.8, 1.1, 1.4, 1.7, 2.0m.  

The experiment was carried out on planet Earth, under the condition of which the friction bar was slided to half-way. The pendulum bob was skewed off at a right angle which was 90 and was let to swing. Simultaneously, the digital clock was set to calculate the time for 10 cycles. The results then were recorded individually for each length and were written down in the form of a table at the end of the experiment. Subsequently, the figure was being analysed in order to determine the value of gravitational acceleration ‘g’on Earth. The uniform method was used to carry out the value of g on planet X.

Figure 1. The arrangment of apparatus (Tutorvista, n.d.)[3]

1.790

3.204

1.1

21.11

2.111

4.456

1.4

23.66

2.366

5.598

1.7

26.19

2.619

6.859

2.0

28.37

2.837

8.049

Table 1. Data collected and analysed from the experiment on planet Earth

The periodT2increased steadily with respect to the length l (Graph 1). According to the graph, at 0.5m long, the square of the cycle was 2.022s. At the length of 2.0m, it increased and reached a peak of 8.049s.

Gradient trend line = 4.02

Gradient max = 4.04

Gradient min = 4.01

• Estimation of errors:
  
  

planet Earth. It could be concerned that the gravitational pull-down force on planet X is greater than the one on Earth.

The value of R2is also close to 1, which implies the trend line is very linear. In the same way, the experiment was done smoothly and carried out the precise results. Since the investigation was performed online, the jeopardies such as the effect of wind, the uncontrollable amount of friction and mistakes in length measurement have been eliminated. The anomalous errors are mainly due to the slow reaction of timing the clock. Hence, an automatic digital clock should be used in the future experiment to improve more the precision.

[1] Tutorvista, n.d. Simple pendulum[online] Available at: <http://www.tutorvista.com/content/physics/

physics-i/measurement-and-experimentation/simple-pendulum.php?laws-of-simple-pendulum>[Accesses 29th February 2012]

[2] PhET, 2011. Pendulum lab 2.03 [online] Available at: <http://phet.colorado.edu/sims/pendulum-lab/pendulum-lab_en.html> [Accessed 25th February 2012]

[3] Tutorvista, n.d. [image online] Available at: <http://images.tutorvista.com/content/measurement-and-experimentation/time-period-measurement.jpeg> [Accessed 28th February 2012]

This student written piece of work is one of many that can be found in our University Degree Physics section.