Exam 1 Phys

Below is a concise “cheat sheet” covering the key ideas, equations, and solution approaches for a first test on Simple Harmonic Motion (SHM), damped/forced oscillations, resonance, and coupled oscillators. It’s organized by topic so you can quickly reference important formulas and concepts.

1. Simple Harmonic Motion (SHM)

1.1. Definition and Conditions

• SHM occurs when the restoring force is proportional (and opposite in direction) to the displacement: F=−kx⟹mx¨=−kx.F = -kx \quad \Longrightarrow \quad m \ddot{x} = -kx.

• Angular frequency: ω0=km\omega_0 = \sqrt{\tfrac{k}{m}}.

1.2. Equations of Motion

1. Force/Acceleration form: x¨+ω02 x=0.\ddot{x} + \omega_0^2\,x = 0.

2. General solution: x(t)=Acos⁡(ω0t)+Bsin⁡(ω0t)orx(t)=Ccos⁡(ω0t−ϕ).x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) \quad\text{or}\quad x(t) = C\cos(\omega_0 t - \phi).

1.3. Energy in Undamped SHM

• Kinetic energy: K=12mx˙2K = \tfrac{1}{2} m \dot{x}^2.

• Potential energy: U=12kx2U = \tfrac{1}{2} k x^2.

• Total energy: Etotal=K+U=12kA2=constant (for undamped motion).E_\text{total} = K + U = \tfrac{1}{2} k A^2 = \text{constant (for undamped motion)}.

1.4. Phasor Representation

• Phasor: Treat cos⁡(ωt)\cos(\omega t) as the real part of e iωt e^{\,i\omega t}.

• Useful for adding or comparing oscillatory terms—especially in forced or coupled systems.

2. Damped SHM

2.1. Equation of Motion

• If a damping force is proportional to velocity, Fdamp=−b x˙F_\text{damp}=-b\,\dot{x}, we get: mx¨+bx˙+kx=0⟹x¨+2ζω0 x˙+ω02 x=0, m \ddot{x} + b \dot{x} + k x = 0 \quad\Longrightarrow\quad \ddot{x} + 2\zeta\omega_0\,\dot{x} + \omega_0^2\,x = 0, where ζ=b2mω0(damping ratio). \zeta = \frac{b}{2m\omega_0} \quad\text{(damping ratio)}.

2.2. Types of Damping

1. Under-damped (ζ<1\zeta < 1): Oscillations occur with an exponentially decaying amplitude.

2. Critically damped (ζ=1\zeta = 1): System returns to equilibrium as quickly as possible without oscillating.

3. Over-damped (ζ>1\zeta > 1): Returns to equilibrium slowly, no oscillations.

2.3. Under-Damped Solution

• For under-damping (ζ<1\zeta < 1), the solution is often written: x(t)=A e−ζω0 tcos⁡(ωd t−ϕ),whereωd=ω01−ζ2. x(t) = A\,e^{-\zeta\omega_0\,t}\cos(\omega_d\,t - \phi), \quad\text{where}\quad \omega_d = \omega_0\sqrt{1-\zeta^2}.

3. Driven (Forced) SHM

3.1. Equation of Motion

• A driving force Fdrive=F0cos⁡(ω t) F_\text{drive} = F_0 \cos(\omega\,t) adds to the damped equation: mx¨+bx˙+kx=F0cos⁡(ωt), m \ddot{x} + b \dot{x} + k x = F_0\cos(\omega t), or in normalized form: x¨+2ζω0 x˙+ω02 x=F0mcos⁡(ωt). \ddot{x} + 2\zeta\omega_0\,\dot{x} + \omega_0^2\,x = \frac{F_0}{m}\cos(\omega t).

3.2. Steady-State Solution & Amplitude

• The long-term (steady-state) motion oscillates at the driving frequency ω\omega.

• Steady-state amplitude X(ω)X(\omega) is given by (for under-damped systems): X(ω)=F0/m(ω02−ω2)2+(2ζω0ω)2. X(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}}.

• Resonance occurs near ω≈ωr=ω01−2ζ2\omega \approx \omega_r = \omega_0 \sqrt{1 - 2\zeta^2} for small damping (ζ≪1\zeta \ll 1).

4. Quality Factor (Q)

• A measure of the “sharpness” of resonance and how quickly energy is lost: Q=ω02ζ ω0  =  12ζorQ=mω0b. Q = \frac{\omega_0}{2\zeta\,\omega_0} \;=\; \frac{1}{2\zeta} \quad \text{or} \quad Q = \frac{m\omega_0}{b}.

• High QQ ⇒ system rings longer (light damping).

5. Energy Considerations in SHM

• Undamped: Total energy is constant and shared periodically between potential (12kx2\tfrac{1}{2}kx^2) and kinetic (12mx˙2\tfrac{1}{2}m\dot{x}^2).

• Damped: Energy exponentially decreases over time due to damping.

• Driven: Steady-state balance between energy input from the driving force and the energy lost to damping.

6. Coupled Oscillators and Normal Modes

6.1. Basic Idea

• When two masses (or oscillators) are connected (e.g., two masses and three springs), each mass can influence the other’s motion.

• The system can often be described by simultaneous second-order ODEs.

6.2. Normal Modes

• Normal modes are independent oscillation patterns in which the entire system moves with a single frequency.

• For two coupled masses, you get:

  1. Symmetric mode (both masses in phase),

  2. Antisymmetric mode (masses out of phase).

6.3. General Motion

• Any initial motion can be expressed as a linear combination of the normal modes.

• Mathematically, you find normal mode frequencies by solving the determinant of the coupled system’s coefficients = 0.

7. Practical Approaches to Problems

1. Identify the type of motion (undamped SHM, damped SHM, forced/driven, or coupled).

2. Use the correct differential equation:

   • x¨+ω02x=0\ddot{x} + \omega_0^2 x = 0 (pure SHM),

   • x¨+2ζω0x˙+ω02x=0\ddot{x} + 2\zeta \omega_0 \dot{x} + \omega_0^2 x = 0 (damped),

   • x¨+2ζω0x˙+ω02x=F0mcos⁡(ωt)\ddot{x} + 2\zeta \omega_0 \dot{x} + \omega_0^2 x = \tfrac{F_0}{m}\cos(\omega t) (driven).

3. Solve or recall standard solutions:

   • For pure SHM: x(t)=Acos⁡(ω0t)+Bsin⁡(ω0t)x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t).

   • For damped or forced: Use exponential ansatz eαte^{\alpha t} or particular solutions for the driving.

4. Check boundary/initial conditions (e.g., x(0)x(0), x˙(0)\dot{x}(0)) to find integration constants A,B,ϕA, B, \phi.

5. For energy: Write out KK and UU, then sum them. Look for constancy or exponential decay.

6. For resonance: Look at amplitude vs. driving frequency; identify ωr\omega_r.

7. For coupled oscillators: Write down the set of equations, assume a mode shape x1=X1eiωtx_1 = X_1 e^{i\omega t}, x2=X2eiωtx_2 = X_2 e^{i\omega t}, solve for possible ω\omega values (the normal modes).

8. Useful Formulas at a Glance

1. Undamped natural frequency: ω0=km. \omega_0 = \sqrt{\frac{k}{m}}.

2. Damped frequency (under-damped): ωd=ω01−ζ2. \omega_d = \omega_0\sqrt{1-\zeta^2}.

3. Damping ratio: ζ=b2km. \zeta = \frac{b}{2\sqrt{km}}.

4. Quality factor: Q=ω02ζ ω0=12ζ. Q = \frac{\omega_0}{2\zeta\,\omega_0} = \frac{1}{2\zeta}.

5. Forced steady-state amplitude: X(ω)=F0/m(ω02−ω2)2+(2ζω0ω)2. X(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}}.

6. Resonance frequency (small damping): ωr≈ω01−2ζ2. \omega_r \approx \omega_0\sqrt{1 - 2\zeta^2}.

Final Tips

• Keep track of phases: In phasor diagrams or solutions with sines/cosines, the phase ϕ\phi is often as important as the amplitude.

• Graphical analysis:

  • Plot x,v,ax, v, a vs. time to see how they are out of phase (especially in standard SHM where vv leads xx by π/2\pi/2, etc.).

  • Plot K,U,K, U, and EtotalE_\text{total} to observe energy exchange or decay.

• Dimensionless form: Many problems use dimensionless damping ratio ζ\zeta, or measure frequency relative to ω0\omega_0, so watch your units carefully.

• Coupled systems: Always reduce to the normal mode picture if possible—sum and difference coordinates can simplify the analysis.

Use this overview to quickly recall the main equations and solution methods during your exam preparation. Good luck!