Answer to Question #100965 in Physics for Emmanuel

Question

Transform the equation T = 2π√((h-s)/g) into linear form so that S becomes the subject of the formula. (Note: T and S are variables)
Write down the theoretical slope and intercept
Describe how the acceleration due to gravity, g can be deduced
State two sources of error and precautions that can be taken during this experiment to minimize the errors

Answer 1

Fig.1

The true equation

(1) $t=\sqrt{\frac{2(h-s)}{g}}$

is equation for time of free fall motion from height $h$ to height $s$. To find $s$ we square both sides of the formula, multiply by $g$ and divide by 2 we get: $h-s=\frac{gt^2}{2}$ . Then move $h$ to the right side and change the sign in both parts. Finally we get

(2)$s(t)=h-\frac{gt^2}{2}$ , [m]

The graph of this function is shown in Fig. 1 ,

and the graph of the change in speed with time in Fig. 2.

Fig.2

The first who scientific studied speed and velocity, gravity and free fall was Galileo in 1564-1642 years. If we know the height $h$ at which the body is released into free fall and the time $t$ during which it reaches the earth $(s=0)$, we can calculate the acceleration with which it moved. From (2) this is

(3) $g=2\cdot \frac{h}{t^2}$

In the time of Galileo, all philosophers were sure that all bodies fall with different accelerations. Galileo's merit lies in the fact that he understood the role of air resistance in the fall of bodies of different sizes and of specific gravity, etc. and showed that in a pipe with evacuated air, bird feathers and lead weights fall with the same acceleration $g\backsimeq 9.8 m s^{-2}$ .

Having pumped out air, he removed the main source of error in determining the acceleration of gravity. For accurate measurement, it was necessary to create an accurate watch - a stopwatch capable of measuring small (fractions of a second) time segments. This is determined by the fact that even from the highest towers, the fall of bodies occurs very quickly (as can be seen from figure 1 for $h=10m; t<1.5s$ ).