Measuring g using an electronic timer

In this method, a steel ball-bearing is held by an electromagnet (Figure 2.23). When the current to the magnet is switched off, the ball begins to fall and an electronic timer starts. The ball falls through a trapdoor, and this breaks a circuit to stop the timer. This tells us the time taken for the ball to fall from rest through the distance h between the bottom of the ball and the trapdoor.
Here is how we can use one of the equations of motion to find g:
displacement $s = h$
time taken $= t$
initial velocity $u = 0$
acceleration $a = g$
Substituting in $s = ut + \frac{1}{2}a{t^2}$ gives:
$h = \frac{1}{2}g{t^2}$
and for any values of h and t we can calculate a value for g.

A more satisfactory procedure is to take measurements of t for several different values of h. The height of the ball bearing above the trapdoor is varied systematically, and the time of fall measured several times to calculate an average for each height. Table 2.4 and Figure 2.24 show some typical results. We can deduce g from the gradient of the graph of h against ${t^2}$.
The equation for a straight line through the origin is:

$y = mx$

In our experiment we have:

The gradient of the straight line of a graph of h against ${t^2}$ is equal to $\frac{g}{2}$.

Therefore:

$\begin{array}{l}
gradient = \frac{g}{2}\\
 = \frac{{0.84}}{{0.20}}\\
 = 4.2\\
g = 4.2 \times 2 = 8.4\,m\,{s^{ - 2}}
\end{array}$

Sources of uncertainty

The electromagnet may retain some magnetism when it is switched off, and this may tend to slow the ball’s fall. Consequently, the time t recorded by the timer may be longer than if the ball were to fall completely freely. From $h = \frac{1}{2}g{t^2}$ , it follows that, if t is too great, the experimental value of g will be too small. This is an example of a systematic error – all the results are systematically distorted so that they are too great (or too small) as a consequence of the experimental design.
Measuring the height h is awkward. You can probably only find the value of h to within $ \pm 1mm$ at best.
So there is a random error in the value of h, and this will result in a slight scatter of the points on the graph, and a degree of uncertainty in the final value of g.
If you just have one value for h and the corresponding value for t you can use the uncertainty in h and t to find the uncertainty in g.
The percentage uncertainty in g = percentage uncertainty in $h + 2 \times $ percentage uncertainty in t.
For more about errors and combining uncertainties, see Chapter ${P_1}$.

Measuring g using a ticker-timer

Figure 2.25: shows a weight falling. As it falls, it pulls a tape through a ticker-timer. The spacing of the dots on the tape increases steadily, showing that the weight is accelerating. You can analyse the tape to 
find the acceleration, as discussed in Practical Activity 2.1.

This is not a very satisfactory method of measuring g. The main problem arises from friction between the tape and the ticker-timer. This slows the fall of the weight and so its acceleration is less than g. (This is another example of a systematic error.)
The effect of friction is less of a problem for a large weight, which falls more freely. If measurements are made for increasing weights, the value of acceleration gets closer and closer to the true value of g.

Measuring g using a light gate

Figure 2.26: shows how a weight can be attached to a card ‘interrupt’. The card is designed to break the light beam twice as the weight falls. The computer can then calculate the velocity of the weight twice as it falls, and hence find its acceleration:

$initial\,velocity\,u = \frac{x}{{{t_2} - {t_1}}}$
$final\,velocity\,v = \frac{x}{{{t_4} - {t_3}}}$

Therefore:

$acceleration\,a = \frac{{v - u}}{{{t_3} - {t_1}}}$

The weight can be dropped from different heights above the light gate. This allows you to find out whether its acceleration is the same at different points in its fall. This is an advantage over Method 1, which can only measure the acceleration from a stationary start.