https://www.physik.uni-leipzig.de/jvwikis/mechanics/ 2026-05-01T07:11:56+00:00 https://www.physik.uni-leipzig.de/jvwikis/mechanics/ https://www.physik.uni-leipzig.de/jvwikis/mechanics/_media/wiki/favicon.ico text/html 2022-01-04T03:15:32+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.1_motivation_and_outline?rev=1641262532&do=diff 5.1 Motivation and outline: How do particles collide? In order to get a first impression about this idea we consider the case of two particles at the positions $\mathbf q_i$, $i \in \{1,2\}$ that interact by a repulsive Coulomb force that derives from a potential $\Phi_C(\left\lvert \mathbf R \right\rvert)$ with $\mathbf R = \mathbf q_2 - \mathbf q_1$, \begin{align*} \Phi_C (\left\lvert \mathbf R \right\rvert) = \frac{ C }{ \left\lvert \mathbf R \right\rvert } \quad \Rightarrow \quad … text/html 2022-01-04T05:12:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.2_collisions_of_particles?rev=1641269544&do=diff 5.2 Collisions of hard-ball particles We consider two spherical particles and denote their radii and masses as $R_i$ and $m_i$ with $i \in \{1,2\}$, respectively. At the initial time $t=t_0$ the particles motion is not (yet) subjected to a force such that \begin{align*} \mathbf q_i (t) = \mathbf q_i ( t_0 ) + v_i \: ( t - t_0 ) \, , \quad\text{for}\quad i \in \{1,2\} \end{align*} 5.2.1 Center of mass motion $\mathbf Q(t)$$\mathbf r (t)$$M = m_1 + m_2$\begin{align} M \; \mathbf Q (t) … text/html 2022-01-04T20:29:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.3_volume_integrals?rev=1641324566&do=diff 5.3 Volume integrals — A professor falling through Earth The center of mass of a set of particles was defined in Equation 4.6.1 as a weighted sum of their positions. Now we consider an extended object that is characterized by a mass distribution $\rho(\mathbf q)$. We will always assume that the distribution varies slowly in space in side the object. Outside it vanishes. The weighted sum over the particle positions will then be generalized to become a volume integral. $\{ s_i \}$$R \subset \ma… text/html 2022-01-06T05:22:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.4_center_of_mass_and_spin?rev=1641442942&do=diff 5.4 Center of mass and spin of extended objects We consider a setting where there are only long distance force like gravity and no collisions between objects. The explicit calculation for the case of gravity in the previous section entails that in such a setting the force exerted by a planet on a point particle is identical to the one exerted by a mass point of identical mass that is located at the center of the planet (see also $\mathbf q$$\hat{\boldsymbol e}_1(t), \dots , \hat{\boldsymbol… text/html 2022-01-07T21:33:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.5_bodies_with_internal_degrees_of_freedom?rev=1641587614&do=diff 5.5 Bodies with internal degrees of freedom: Revisiting mobiles In Section 2.10 we worked out the positions of masses for a mobile where all masses are the same and where all sticks are straight. It is worth while to revisit this problem from a more advanced mathematical perspective.$\dot{\boldsymbol q} = \mathbf 0$$\mathbf F_{\text{tot}}$$\mathbf F_s$$M\, \mathbf g$\begin{align*} \mathbf 0 = \mathbf F_{\text{tot}} = \mathbf F_s + M\, \mathbf g \quad \Rightarrow \quad \mathbf F_s = -… text/html 2022-01-07T21:37:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.6_worked_example?rev=1641587874&do=diff 5.6 Worked example: Reflection of balls We consider the reflection of a ball from the ground, the lower side of a table, and back. The ball is considered to be a sphere with radius $R$, mass $m$, and moments of inertia $m \alpha R^2$ (by symmetry they all agree). Its velocity at time $t_0$$\dot{\mathbf z}_0$$\mathbf\omega_0 = \mathbf 0$$n^{\textrm{th}}$$\dot{\mathbf z}_n$$\mathbf\omega_n$$x$$x$$z$$\mathbf F_{\perp}$$\mathbf F_{\parallel}$$\mathbf F_{\perp}$$v_n = \hat x \cdot \mathbf{\dot z}$\… text/html 2022-02-01T23:40:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/5.7_problems?rev=1643755224&do=diff 5.7 Problems 5.7.1 Practicing Concepts Problem 5.11: Determining the volume, the mass, and the center of mass Determine the mass $M$, the area or volume $V$, and the center of mass $\mathbf Q$ of bodies with the following mass density and shape. * A triangle in two dimensions with constant mass density $\rho = 1 \, \text{kg/m$$}$$6 \, \text{cm}$$8 \, \text{cm}$$10 \, \text{cm}$$(a,b)$$R = 5 \, \text{cm}$$\rho = 1 \, \text{kg/m$$}$$M$$V$$\mathbf Q$$0 \leq x \leq W$$0 \leq y \leq B$$\r… text/html 2022-01-02T15:50:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap5/spatial-extension?rev=1641135001&do=diff 5. Impact of Spatial Extension Figure 5.1: Impact of a laser pulse on a microdrop of opaque liquid that is thus blown up; Klein, et al, Phys. Rev. Appl. 3 (2015) 044018 Figure 5.2: Girl playing with clackers. Punt/Anefo, Amsterdam 1971, CC0 Figure 5.3: Man running to return a tennis ball. Charlie Cowins from Belmont, NC, USA, CC by 2.0 In Chapter 4 we discussed the motion of point particles. However, in our environment the spatial extension of particles in crucial. Physical objects a…