Physics, 13.12.2020 15:00 NEUROPHARMACOLOGICAL

Un pendule est constitue par une masse ponctuelle m= 0,1kg accrocher a un fil sans masse de longueur L = 0,4 m on ecarte ce pendule d, un angle Om= 0,1 rad de sa position d’equilibre et on l’abondonne sans vitesse initiale a la date t0=0. Prendre g= 10 m/s2 et on neglige la force de frottement a-) Etablir l’equation defferentielle de mouvement

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Answer from: babycakes08d

$0.4\,\frac{d^2\theta}{dt^2} =-10\,\,\theta$

Explanation:

Notice that the pendulum's mass is 0.1 kg, the length of the string is 9.4 m, the starting angle is 0.1 radians and it is started without imparting any velocity (just letting the mass drop). therefore, the differential equation for the pendulum's motion can be obtained by decomposing the forces acting on the pendulum into

those along the string (tension and component of the pendulum's weight along the string's direction) and the perpendicular to such (tangential to the pendulum's motion. Notice that the Net Force in the system is that associated with the tangential component of the weight (see attached image)

$F_{net}=-m\,g\,sin(\theta)$

For small angles (which is our case for the angle of 0.1 radians), $sin(\theta)$ can be approximated by the first term of this function’s Taylor expansion: $\theta$ , thus leading to the equation:

$F_{net}=-m\,g\,\theta\\m\,*\,a_t=-m\,g\,\theta\\m\,*\,\frac{d\,v_t}{dt} =-m\,g\,\theta$

which dividing both sides by the pendulum's mass, and replacing the tangential velocity with the angular velocity times radius of motion (length of pendulum), becomes:

$\frac{d\,v_t}{dt} =-\,g\,\theta\\\frac{d\,\omega\,L}{dt} =-\,g\,\theta\\L\frac{d^2\theta}{dt^2} =-g\,\theta$

And for our given values, this becomes:

$0.4\,\frac{d^2\theta}{dt^2} =-10\,\,\theta$

Un pendule est constitue par une masse ponctuelle m= 0,1kg accrocher a un fil sans masse de longueur

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Answer from: Quest

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