Physics examples

Example (1): Possible applications of simple pendulum formula:

In early classical mechanics, there is an empirical formula for a simple gravitational pendulum which is made of mass-less string of length L [m] attached smoothly to a heavy bob at the lower end and to a friction-less rigid pivot at the upper end. Entire mechanism is in a uniform gravity field of strength g [N/kg]. Air resistance is neglected.  In equilibrium state, the string is in vertical line. When given an initial push and released, the pendulum swings back and forth at an  amplitude (maximum angle made by string from vertical line). The string remains straight during the swings. The period T [second] of the swing is the time taken for the bob to make a complete cycle.  

Dependencies:

The g-value does not depend on L, T and the angle of swing. it depends on the gravitational field at the place of the experiment.  If the gravitational field is time independent, we may assume that g depends on the location. The maximum angle of swing, if smaller than 1 radian, is assumed to have no influence on the relation between T, L and g. In this case,  

T is a function of L and g. By changing the length L of the string or by moving the experiment to another place where g-value is different, T-value will also change. 

When a (T, g) value is given, the corresponding string length L-value can be calculated from the formula. Practically, it seems impossible for us on earth to perform the experiment with any pre-selected (T, g) value, where T > 0 and g > 0, to measure the corresponding L-value. However, for any positive number x we choose, there exists a place in universe where g-value is x. So, we can do thought experiment to find the observed L-value for any given (T, g) value at that unknown location in universe. That is, L can be treated as a function of (T, g). What we can do is that knowing the local g-value, we set T-value to determine the required L-value to build a simple pendulum with period T.

As well, g can be treated as a function of (T, L) not because g depends on T and L but to indirectly measure the g-value of the location. Suppose we are at a place on earth where  the local g-value is unknown to us. We perform the simple pendulum experiment with a chosen L-value. T-value is measured from the experiment. Using these L and T values, g of the location can be calculated. 

Simple pendulum formula can berewritten as:

Conditions of the formula:

1. Mass of string is negligible.

2. Air resistance is negligible.

3. Gravity field is uniform.

4. L is the length from pivot to the centre of mass of the bob. Mass of bob does not matter.

5. Maximum swing angle is smaller than 1 radian.

Possible applications:

1. Given L and g values, do the experiment and measure the period T. Compare this observed T-value with the calculated T-value and investigate the validity and possible improvements of the experiment.

2. Given the local g-value and any chosen T-value, calculate the required L-value. Use this L-value to build a simple pendulum.  Test the pendulum to check the difference between observed T-value and calculated T-value for improvement.

3. Assuming the local g-value is unknown, do experiments with the simple pendulum with known L-value. Measure the T-value and calculate the g-value. This will be an estimate of the local g-value.

4. Square-root calculator: Suppose we have a precise value of local g, and we have calculated the value of y = √g/(2𝜋) to a good accuracy. Then, to calculate the square-root of a positive number x, we adjust the simple pendulum to have L = x. Swing the pendulum for n number of times. Each time we measure the period T. Get the average T value and multiply it with y for the square-root of x.

Example 1: applications of a formula Example 2: Dependency relations