Physique, lisible pour tous
Equa Diff

Système masse-ressort

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><msub><mi>F</mi> <mi>x</mi></msub><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\sum F_{x} = m a_{x}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>kx</mi><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex"> -kx = m a_{x}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>kx</mi><mo>=</mo><mi>m</mi><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> -kx = m \frac{d^2 x}{dt^2}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>k</mi><mi>m</mi></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{d^2 x}{dt^2} = -\frac{k}{m}x </annotation></semantics></math>$

On vérifie qu’une solution de cette équation est $<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)=A\sin(\omega t + \phi)</annotation></semantics></math>$ avec $<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline"><semantics><mrow><msup><mi>ω</mi> <mn>2</mn></msup><mo>=</mo><mi>k</mi><mo stretchy="false">/</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\omega^{2} = k/m</annotation></semantics></math>$

Le pendule simple

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><msub><mi>F</mi> <mi>y</mi></msub><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\sum F_{y} = m a_{y}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><msub><mi>T</mi> <mi>y</mi></msub><mo>−</mo><mi>mg</mi><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>y</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> T_{y}- mg = m a_{y} = 0</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mi>T</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>mg</mi></mrow><annotation encoding="application/x-tex"> T\cos(\theta) = mg</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mi>T</mi><mo>=</mo><mfrac><mi>mg</mi><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> T = \frac{mg}{\cos(\theta)}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><msub><mi>T</mi> <mi>x</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">T_{x} = -T\sin(\theta) = m a_{x}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>mg</mi><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>sin</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>mg</mi><mspace width="thinmathspace"/><mi>tan</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><msub><mi>a</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">-\frac{mg}{\cos(\theta)}\sin(\theta) = -mg \, \tan(\theta) = m a_{x}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>g</mi><mi>θ</mi><mo>=</mo><msub><mi>a</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">-g\theta = a_{x}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>g</mi><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">-g\theta(t) = \frac{d^2 x(t)}{dt^2}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>g</mi><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">[</mo><mi>θ</mi><mi>L</mi><mo stretchy="false">]</mo></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">-g\theta(t) = \frac{d^2 [\theta L]}{dt^2}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>g</mi><mi>L</mi></mfrac><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">-\frac{g}{L} \theta(t) = \frac{d^2 \theta(t)}{dt^2}</annotation></semantics></math>$

On vérifie qu’une solution de cette équation est $<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline"><semantics><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>θ</mi> <mi>max</mi></msub><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta(t)=\theta_{max}\sin(\omega t + \phi)</annotation></semantics></math>$ avec $<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline"><semantics><mrow><msup><mi>ω</mi> <mn>2</mn></msup><mo>=</mo><mi>g</mi><mo stretchy="false">/</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\omega^{2} = g/L</annotation></semantics></math>$

Revised on February 20, 2022 17:18:10 by Anonymous Coward

Physique, lisible pour tous Equa Diff

Système masse-ressort

Le pendule simple

Physique, lisible pour tous
Equa Diff