When dealing with a Lagrangian mechanics problem, follow this structured approach to ensure success:
ẍ=−M+mmcosαẌx double dot equals negative the fraction with numerator cap M plus m and denominator m cosine alpha end-fraction cap X double dot Substitute into the -equation to find the wedge acceleration Ẍcap X double dot
From ( \dot X = - \fracm\cos\alphaM+m,\dot x ), differentiate: [ \ddot X = - \fracm\cos\alphaM+m,\ddot x ] Substitute into the ( x )-equation: [ m\left( -\fracm\cos\alphaM+m,\ddot x \cos\alpha + \ddot x \right) = m g \sin\alpha ] [ \ddot x \left( 1 - \fracm\cos^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x \left( \fracM+m - m\cos^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x \left( \fracM + m\sin^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x = \frac(M+m)g\sin\alphaM + m\sin^2\alpha ] Then: [ \ddot X = - \fracm\cos\alphaM+m \cdot \frac(M+m)g\sin\alphaM + m\sin^2\alpha ] [ \boxed\ddot X = - \fracm g \sin\alpha \cos\alphaM + m\sin^2\alpha ]
V=mgz=−mgRcosθcap V equals m g z equals negative m g cap R cosine theta
When dealing with a Lagrangian mechanics problem, follow this structured approach to ensure success:
ẍ=−M+mmcosαẌx double dot equals negative the fraction with numerator cap M plus m and denominator m cosine alpha end-fraction cap X double dot Substitute into the -equation to find the wedge acceleration Ẍcap X double dot
From ( \dot X = - \fracm\cos\alphaM+m,\dot x ), differentiate: [ \ddot X = - \fracm\cos\alphaM+m,\ddot x ] Substitute into the ( x )-equation: [ m\left( -\fracm\cos\alphaM+m,\ddot x \cos\alpha + \ddot x \right) = m g \sin\alpha ] [ \ddot x \left( 1 - \fracm\cos^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x \left( \fracM+m - m\cos^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x \left( \fracM + m\sin^2\alphaM+m \right) = g \sin\alpha ] [ \ddot x = \frac(M+m)g\sin\alphaM + m\sin^2\alpha ] Then: [ \ddot X = - \fracm\cos\alphaM+m \cdot \frac(M+m)g\sin\alphaM + m\sin^2\alpha ] [ \boxed\ddot X = - \fracm g \sin\alpha \cos\alphaM + m\sin^2\alpha ]
V=mgz=−mgRcosθcap V equals m g z equals negative m g cap R cosine theta
