A symmetric top ((I_1=I_2\neq I_3)) with no torque. Show that (\omega_3) constant, and (\boldsymbol\omega) precesses around symmetry axis.
Use angular momentum conservation (L = mr^2\dot\theta) and energy: [ E = \frac12m\dotr^2 + \fracL^22mr^2 - \frackr ] Set (u = 1/r), get Binet’s equation: [ \fracd^2ud\theta^2 + u = -\fracmL^2 u^2 F(1/u) ] For inverse-square law, solution: (u = \fracmkL^2 + A\cos(\theta - \theta_0)), i.e., conic sections. Chapter 5: Lagrangian Formulation Core concepts: Hamilton’s principle, generalized coordinates, Lagrange’s equations, constraints, cyclic coordinates.
Solve ( \ddotx + 2\beta \dotx + \omega_0^2 x = (F_0/m)\cos\omega t ) via complex exponentials: assume (x = \textRe[A e^i\omega t]), substitute to get [ A = \fracF_0/m\omega_0^2 - \omega^2 + 2i\beta\omega ] Amplitude ( |A| = \fracF_0/m\sqrt(\omega_0^2 - \omega^2)^2 + 4\beta^2\omega^2 ). Chapter 4: Gravitation and Central Forces Core concepts: Reduced mass, effective potential, orbits, Kepler’s laws, scattering. symon mechanics solutions pdf
In rotating Earth frame: ( \mathbfa \textrot = \mathbfa \textinertial - 2\boldsymbol\omega \times \mathbfv_\textrot - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbfr) ). Neglect centrifugal for short-range. For vertical motion, Coriolis gives eastward acceleration: (a_x = 2\omega v_z \cos\lambda). Integrate twice. Chapter 8: Rigid Body Dynamics Core concepts: Inertia tensor, principal axes, Euler’s equations, torque-free precession.
A particle of mass (m) moves under central force (F(r) = -k/r^2). Derive the orbit equation. A symmetric top ((I_1=I_2\neq I_3)) with no torque
A mass (m) on a spring (k) with damping (b) and driving force (F_0 \cos \omega t). Find steady-state amplitude and phase.
Two masses (m_1, m_2) coupled by springs (k_1, k_2, k_3). Find normal modes. In rotating Earth frame: ( \mathbfa \textrot =
String fixed at both ends, initial displacement (f(x)), initial velocity zero. Find subsequent motion.
