Oscillations — MHT-CET Physics

Formulas (3)

• Velocity, Acceleration and Displacement · SHM velocity
  $$v = \omega\sqrt{A^2 - x^2}, \qquad a = -\omega^2 x$$
• Phase, and Time to Reach a Point · Phase
  $$x = A\sin(\omega t + \alpha), \qquad \omega = \frac{2\pi}{T}$$
• Restoring Forces, Detachment and Damping · Damped amplitude
  $$A = A_0e^{-\lambda t}, \qquad \omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}$$

Watch out for (3)

• Subtracting the accelerations instead of adding
  → Velocity, Acceleration and Displacement
• Using cos when the motion starts at the mean
  → Phase, and Time to Reach a Point
• Thinking damping only shrinks the amplitude
  → Restoring Forces, Detachment and Damping

Formulas (3)

• The Spring-Mass Period · Spring-mass
  $$T = 2\pi\sqrt{\frac{m}{k}}$$
• Springs in Series, in Parallel, and Cut · Combining springs
  $$k_{\parallel} = k_1 + k_2, \qquad \frac{1}{k_{\text{series}}} = \frac{1}{k_1} + \frac{1}{k_2}$$
• Other Oscillators: Bowls, Floating Blocks, Liquid Columns, Magnets · Liquid column and floating block
  $$T_{\text{column}} = 2\pi\sqrt{\frac{M}{2Adg}}, \qquad T_{\text{block}} = 2\pi\sqrt{\frac{ad}{g}}$$

Watch out for (4)

• Scaling the period with the mass itself
  → The Spring-Mass Period
• Putting g into a spring's period
  → The Spring-Mass Period
• Treating springs on opposite sides as series
  → Springs in Series, in Parallel, and Cut
• Using Adg for a liquid column
  → Other Oscillators: Bowls, Floating Blocks, Liquid Columns, Magnets

Formulas (2)

• Kinetic and Potential Energy at a Point · Energy split
  $$\frac{U}{E} = \frac{x^2}{A^2}, \qquad \frac{K}{E} = 1 - \frac{x^2}{A^2}$$
• Total Energy and What It Depends On · Total energy
  $$E = \tfrac{1}{2}m\omega^2A^2$$

Watch out for (2)

• Putting the energies equal at half the amplitude
  → Kinetic and Potential Energy at a Point
• Forgetting the frequency in the energy
  → Total Energy and What It Depends On

Formulas (3)

• Period and Length · Simple pendulum
  $$T = 2\pi\sqrt{\frac{L}{g}}$$
• Effective g: Lifts, Moving Supports and Orbit · Accelerating support
  $$T' = 2\pi\sqrt{\frac{L}{g \pm a}}$$
• Speed and Tension Along the Swing · Tension along the swing
  $$T_{\max} = mg(3 - 2\cos\theta), \qquad T_{\min} = mg\cos\theta$$

Watch out for (3)

• Changing the bob to change the period
  → Period and Length
• Adding the acceleration in the wrong direction
  → Effective g: Lifts, Moving Supports and Orbit
• Taking the minimum tension as mg
  → Speed and Tension Along the Swing