Chapter 3: Problem 2
A particle of mass \(m\) is subject to a force \(\boldsymbol{F}\). Obtain the equations of motion in cylindrical polar coordinates.
Chapter 3: Problem 2
A particle of mass \(m\) is subject to a force \(\boldsymbol{F}\). Obtain the equations of motion in cylindrical polar coordinates.
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Get started for freeA particle of unit mass is constrained to move on the surface of a unit sphere, but is otherwise free. Show that the dynamical trajectories are great circles traversed at uniform speeds. Show that if \(\gamma\) is a complete circuit of a great circle in time \(t\), then $$ J_{L}(v)=2 \pi^{2} / t $$ Does \(\gamma\) minimize \(J_{L}\) over all curves on the sphere that start and end at a point \(P\) on the equator and take time \(t\) for the round trip from \(P\) back to \(P ?\)
In Example \(3.8\), find \(\tilde{Q}_{1}\) and \(\tilde{Q}_{2}\) in terms of the components of \(\boldsymbol{F}\) by considering the work done under suitable small displacements. Check that the same expressions for the \(\tilde{Q}_{a}\) s follow from the chain rule in the case that \(\boldsymbol{F}=-\boldsymbol{\nabla} U\).
A particle of mass \(m\) is free to move in a horizontal plane. It is attached to a fixed point \(O\) by a light elastic string, of natural length \(a\) and modulus \(\lambda\). Show that the tension in the string is conservative and show that the Lagrangian for the motion when \(r>a\) is $$ L=\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)-\frac{\lambda}{2 a}(r-a)^{2} $$ where \(r\) and \(\theta\) are plane polar coordinates with origin \(O\).
A particle of unit mass is subject to an inverse-square-law central force $$ \boldsymbol{F}=-\frac{\boldsymbol{r}}{r^{3}} $$ where \(r=|\boldsymbol{r}|\) and \(\boldsymbol{r}\) is the position vector from the origin of an inertial frame. Show that the motion is governed by the Lagrangian $$ L=\frac{1}{2} \dot{r} \cdot \dot{r}+\frac{1}{r} $$ Write down the equations of motion in spherical polar coordinates and show that there are solutions with \(\theta=\pi / 2\) throughout the motion.
\({ }^{\dagger}\) Two particles \(P_{1}\) and \(P_{2}\) have respective masses \(m_{1}\) and \(m_{2}\) and are attracted to each other by a force with time- independent potential \(U(\boldsymbol{r})\), where \(\boldsymbol{r}\) is the vector from \(P_{1}\) to \(P_{2}\). (1) Show that the motion of the centre of mass is the same as in the case \(U=0\). (2) Show that the motion of \(P_{2}\) relative to \(P_{1}\) is the same as in the case that \(P_{1}\) is fixed and \(P_{2}\) is attracted to \(P_{1}\) by a force with potential $$ V=\frac{m_{1}+m_{2}}{m_{1}} U . $$
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