Theoretical Mechanics Final Resources

“Once more unto the breach”

Topics

The two topics that will be on the final exam are coupled oscillations and vibrations. It may or may not include any other topics.

• See section below on Final Exam Information

2024-03-26 Oscillations

2024-03-29 Coupled Oscillations

2024-04-05 General Coupled Oscillations

Equations

General Mathematical Relations

1. Taylor Series

$$(x+h)=(x){+}^{\prime }(x)h{+}^{\prime \prime }(x)\frac{h^2}{2!}+\dots$$

2. Small Angle Approximation

$$\tan \theta \approx \sin \theta \approx \theta$$ $$\cos \theta \approx 1$$

3. Harmonic Oscillator diffeq

$$\ddot{\theta }=-{\omega }^2\theta$$

From exam 1 topics:

Equations:

1. . Kinetic energy formulas

$$T_{\text{rolling}}=\frac{1}{2}I{\omega }^2$$ $$T_{\text{linear}}=\frac{1}{2}m{v}^2$$

2. Potential Energy formulas

$$U_{\text{ gravity}}=mgh\text{ (depends on coordinate system)}$$ $$U_{\text{spring}}=\frac{1}{2}k\Delta x^2\text{ (hookian spring)}$$

3. Lagrangian

$$L=T-U$$

4. Euler-Lagrange Equation

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$$

Exam 2 Equations on final

1. Modified Lagrangian (optional, can use substitution)

$$\tilde{L}=L-\lambda f$$ $$f=\text{ force }$$

2. Centrifugal Potential

$$U_{cf}=\frac{l}{2\mu r^2}$$

4. Effective Potential

$$U_{eff}=U+U_{cf}$$

5. Continuous Inertia Tensor

$$I_{ij}={\int }_{v}p(\vec{r})\left({\delta }_{ij}\sum _k{x}_k^2-x_i{x}_j\right)dV$$

7. Eigenvalue and Eigenvector Relation

$$(\mathbf{A}-\lambda \mathbf{I})\cdot \vec{\mathbf{v}}=0$$

and

$$\det (\mathbf{A}-\lambda \mathbf{I})=0$$

Final-Exclusive Topics (Study these)

1. $A_{jk}$ Matrices
2. Continuous systems

Final Exam information

6 problems

• Non-Relativistic Newtonian Mechanics Problem 1
  • $F=MA$
  • Free body diagrams
  • Generating newtons second law
  • Rearrange and solve integrals
  • Position and velocity as function of time
  • Drag forces?
• Central Force Problem Problem 2
  • Starting with from Lagrangian
    • See homework 7 problem 3
  • Coordinate transformation to uncouple differential equations
  • Reduced mass equation !!
• Lagrangian w/ constraints Problem 3
  • Not going to be forced to do Lagrange multiplier. Can if you want.
• Moment of inertia problem Problem 4
  • Continuous distribution

$$I_{ij}=\int \rho \left({\delta }_{ij}\sum _k{x}_{\alpha k}^2-x_{\alpha i}{x}_{\alpha j}\right)dV$$

- In Cartesian coordinates

• Coupled Oscillations Problem 5
  • Finding $A_{jk}$ matrices
  • Find determinants to find $\omega$
  • Hw 10
• Continuous Systems: Waves Problem 6
  • Hw 11
  • Plucked and struck strings
  • Maybe its a struck, plucked string
  • Lots of integrals

No Relativity, No Hamiltonian, No Non-inertial reference frame, No scattering

Final Equations list to review

Lagrangian and Euler-Lagrange Equation

$$L=T-U$$ $$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$$

Reduced Mass Equation

$$\frac{m_1{m}_2}{m_1+m_2}=\mu$$

Centrifugal Potential

$$U_{cf}=\frac{l}{2\mu r^2}$$

Effective Potential

$$U_{eff}=U+U_{cf}$$

Continuous Distribution Inertia Tensor

$$I_{ij}=\int \rho \left({\delta }_{ij}\sum _k{x}_{\alpha k}^2-x_{\alpha i}{x}_{\alpha j}\right)dV$$

$A_{jk}$ matrices

$$A_{jk}=\frac{{\partial }^{2}u}{\partial q_j\partial q_k}$$

Eigenfrequency and Eigenvector relations

$$\det [\bar{A}-{\omega }^2\bar{m}]=0$$ $$[\bar{A}-{\omega }^2\bar{m}]\cdot \vec{v}=0$$

${\mu }_r$ formula

$${\mu }_r=\frac{2}{L}\underset{0}{\overset{L}{\int }}q(x,0)\sin \left(\frac{r\pi x}{L}\right)dx$$

${\nu }_r$ formula

$${\nu }_r=-\frac{2}{{\omega }_{r}L}\underset{0}{\overset{L}{\int }}\dot{q}(x,0)\sin \left(\frac{r\pi x}{L}\right)dx$$

position of strings formula

$$q(x,t)=\sum _{r=1}^{\infty }\sin \left(\frac{r\pi x}{L}\right)({\mu }_r\cos ({\omega }_{r}t)-{\nu }_r\sin ({\omega }_{r}t))$$