Leibnitz
Conservation of energy for a pendulum (measurement of centripetal acceleration)
Autor:
Titre 4
Learning objectives :
This activity allows students to experimentally verify the principle of conservation of energy for a pendulum. It links theoretical calculation and practical measurements to validate a fundamental law of physics.
Concepts covered
Energy conservation; Simple pendulum; Centripetal acceleration; Potential and kinetic energy; Periodic movement
What students will do :
The student studies the conservation of mechanical energy of a pendulum using the FizziQ accelerometer. After having established by calculation the theoretical relationship between the initial height and the maximum centripetal acceleration at the low point, the student carries out experimental measurements for different heights and verifies graphically that the centripetal acceleration is indeed proportional to the height.
What is required :
Smartphone with the FizziQ application; Strong string or wire to create a pendulum; Material for fixing the pendulum; Tape measure; Support for measuring different heights; FizziQ experience notebook
Scientific background :
Gottfried Wilhelm Leibniz (1646-1716) was one of the first to formulate the principle of conservation of energy, a fundamental concept of physics. For a simple pendulum, this principle establishes that the total mechanical energy remains constant in the absence of friction. This conservation makes it possible to establish a direct relationship between the starting height and the centripetal acceleration at the lowest point. The theoretical analysis proceeds as follows: 1) At the starting point, at a height h, the energy is entirely potential: E_p = mgh (m: mass, g: gravitational acceleration); 2) At the lowest point, the energy is entirely kinetic: E_c = ½mv² (v: tangential velocity); 3) By conservation of energy: mgh = ½mv², therefore v = √(2gh); 4) The centripetal acceleration is linked to the tangential speed by a_c = v²/r (r: length of the pendulum), which gives: a_c = 2gh/r. This last equation shows that the centripetal acceleration is directly proportional to the initial height h, with a coefficient of proportionality 2g/r. The experiment consists of verifying this relationship by measuring the centripetal acceleration for different starting heights. The smartphone's accelerometer, placed at the end of the pendulum, directly measures this acceleration at the lowest point of the movement. The graphical representation of a_c as a function of h should give a line whose slope allows the ratio 2g/r to be estimated experimentally. Any deviations are mainly explained by friction (air and suspension point) which dissipates part of the energy, particularly for large amplitudes. This experiment perfectly illustrates the predictive power of the principle of conservation of energy, a pillar of classical mechanics.