Free Fall (determination of the standard gravity)

From wwwelab
Revision as of 00:13, 22 March 2012 by Ist165721 (talk | contribs) (Versão inicial)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Description of the Experiment

This experiment will result in the determination of the standard gravity, g.

Experimental Apparatus

This apparatus is a projectile launcher (with an electromagnet) that vertically launches a ping-pong ball from a base.

[fig.1]

The ping-pong ball is launched with a kinetic energy selected by the user (energy applied to the electromagnet) and takes a certain trajectory, afterwards it falls on the base. The height (vertical coordinate) is measured and recorded with an ultrasound sonar, strategically placed above the experiment, that measures the distance between itself and the ball. With the resulting data it is possible to determine g. The ball is sometimes launched with a small horizontal component, but this does not affect measurements.

Since the height is measured by ultrasound, the actual time between samples varies from the one provided by the user: the closer the ball is to the sensor (i.e., higher), the shorter the time between samples is.

Protocol

The user must define the following parameters: launch energy, time between samples and number of samples.

The standard gravity, g, can be established in various ways. The simplest is to fit the mathematical law for the ball's motion, \( x = x_0 + v_0 t + \frac{1}{2}gt^2 \), to the experimental data by choosing three points from the space-time graph and solving the system of equations.

The following graph is a raw data example with \( 20 ms \) between samples.

[fig.2]

This next graph, with the corrected time, the numerical fitting gives the final value \( 9,2 ms^{-2} \) for g.

[fig.3]

Advanced Protocol

Since the ping-pong ball has very low mass, the air resistance will be significant after the ball reaches a certain speed. This way, if we fit the \( x = v_0 t + \frac{1}{2} g t^2 \) to the data right as the motion starts (\(v_0 \simeq 0\)), we can obtain a numerical value for g that is closer to reality. It is interesting to study on how \( g \) changes with speed and observe it's deviation from the expected value \( 9.8 m/s^{-2} \). It can be observed that for smaller trajectories (when the ball bounces), the numerical results are better, or, likewise, in regions where the speed is lower (when the ball reaches a maximum in height).

By considering air friction, the fitting of a more adequate function will result in the value found for \( g \) being more precise and will allow the determination of the air friction constant alpha (\( \alpha \)). If we consider friction to be dependant of \( v^2 \) and applying a force opposite to the movement, we can consider the differential equation:

\[ mg - \alpha v^2 = ma

\] .

Data Analysis

Under construction.

Theoretical Principles

Under construction.

Historical Elements

Under construction.

Bibliography

Under construction.