M easurement procedure and experimental equipment

In this laboratory work, the oscillatory motion of a mathematical pendulum is used to study random distribution and determine gravity acceleration.

M athematical (simple) pendulum is an abstraction – material point suspended on a weightless unstretchable thread. It can be represented as metal sphere (called bob) suspended on a thin thread. The diameter of the bob is much less than the length of a thread; the oscillations are performed in air medium (force of medium resistance may be ignored); the angle of thread deviation about equilibrium position has a small value (3 … 5^o). If such a system is off balance, it will perform free undamped harmonic oscillations.

Pendulum (Fig. 5.4), deviated on an angle  from equilibrium position, performs harmonic oscillations under the influence of restoring force F which is a resultant of gravity force P and thread tension Q.

so ^,

where m is mass of the ball; g is gravity acceleration.

As the sine of a small angle is equal to the value of the angle, we have

^,

where x is displacement of a pendulum with reference to equilibrium position; l is the length of a pendulum thread. Thus

.

The restoring force is tangentially directed to the bob motion trajectory and opposite to the direction of axis x, therefore there is the minus sign. According to the second Newton's law

.

So the pendulum motion equation looks as follows:

or .

The angular (cyclic) natural oscillation frequency _o is defined by the system parameters

.

Thus we have got a differential equation of a mathematical pendulum motion or basic equation of free undamped harmonic oscillations:

.

The general solution of this equation is

,

where A is oscillation amplitude (the greatest displacement of a pendulum equilibrium position); _ot +_o is oscillation phase; _o is initial phase (oscillation phase while t = 0), _o is natural cyclic oscillation frequency (number of oscillations during 2 seconds).

A period of oscillations (time of one complete oscillation) and cyclic frequency are connected by the formula

.

Then the period of mathematical pendulum oscillations is defined by the expression

.

If the time of several oscillations has been measured, the period can be determined by the formula

,

where k is the number of oscillations; t is the time of oscillations.

Using the previous formulas one can get the expression for gravity acceleration magnitude determination

. (5.7)

Work procedure and data processing

Having carried out multiple measurements of quantity t, it is also possible to draw a histogram for measured values and to verify a hypothesis about normal distribution, i.e. to find out, whether the measured values correspond to law of normal distribution described by Gaussian formula.

1. Drawing a histogram.

1.1. Deviate the pendulum by 3^o … 5^o angle and let it off. When the pendulum is in one of the extreme positions, turn on a time measuring device and measure the time of five complete oscillations. Perform the actions 50 times. Enter the results of measurement into the second column of table 5.3.

Table 5.3

Measurement No.	Measured values ti, s	Measured values in ascending order ti, s	Intervals boundary t01 – t02, … t0i – t0k,	Number of values in interval nk
1	2	3	4	5	6
1 2 … … … 49 50

1.2. Analyse the obtained values and arrange them in increasing progression starting from the smallest value t_min up to the largest value t_max. Enter arranged values into the third column of the table.

1.3. Divide the full range of the measured values into 5 equal intervals t and determine boundary values t₀ of each interval in the fourth column (t₀₁=t_min+t; t₀₂=t₀₁+t etc.). Separate the values t_i which are relevant to boundary values by horizontal line in the table.

1.4. Count a number of values n_i which has been included in each interval and enter it into the fifth column. Calculate the rate of reappearance of the measured values n_i/n. Enter the obtained results into the sixth column.

1.4. Draw the histogram of the measured values using as an example the theoretical diagram.

2. Determination of gravity acceleration.

2.1. Find in the histogram the interval with the greatest rate of reappearance of the measured values.

2.2. Calculate the average mean of measured values which are contained within the interval.

2.3. Using expression (5.7) determine the value of gravity acceleration.

2.4. Deduce an equation for calculation of the error of indirect measurement of gravity acceleration.