Oscillations — JEE Mains Physics

Formulas (2)

• Phase, velocity and acceleration from the SHM equation · Displacement, velocity and acceleration in SHM
  $$x = A\sin(\omega t + \phi) \qquad v = A\omega\cos(\omega t + \phi) \qquad a = -\omega^{2}x$$
• Speed at a displacement and amplitude from one snapshot · Speed and acceleration at a displacement
  $$v = \omega\sqrt{A^{2} - x^{2}} \qquad v_{max} = A\omega \qquad a_{max} = \omega^{2}A$$

Watch out for (6)

• At rest means phase π/2, not phase 0
  → Phase, velocity and acceleration from the SHM equation
• The starting direction picks the initial phase
  → Phase, velocity and acceleration from the SHM equation
• Projection on the x-axis is a cosine
  → Phase, velocity and acceleration from the SHM equation
• Divide by the number in front of v² first
  → Speed at a displacement and amplitude from one snapshot
• The v–x graph is an ellipse
  → Speed at a displacement and amplitude from one snapshot
• Acceleration is largest at the extremes
  → Speed at a displacement and amplitude from one snapshot

Formulas (2)

• Time taken to move between two positions in SHM · Time from the phase covered
  $$t = \frac{\Delta\theta}{\omega} = \frac{\Delta\theta}{2\pi}\,T$$
• Adding two SHMs of the same frequency, and telling SHM from periodic motion · Resultant of two SHMs of the same frequency
  $$A = \sqrt{A_1^{2} + A_2^{2} + 2A_1A_2\cos\Delta\phi} \qquad a\sin\omega t + b\cos\omega t = \sqrt{a^{2} + b^{2}}\,\sin\left(\omega t + \tan^{-1}\frac{b}{a}\right)$$

Watch out for (6)

• Starting at an extreme needs the cosine
  → Time taken to move between two positions in SHM
• Equal distances do not take equal times
  → Time taken to move between two positions in SHM
• A quarter period covers A only from the mean or an extreme
  → Time taken to move between two positions in SHM
• Amplitudes add as vectors, not as numbers
  → Adding two SHMs of the same frequency, and telling SHM from periodic motion
• A constant shift does not spoil SHM
  → Adding two SHMs of the same frequency, and telling SHM from periodic motion
• Two different frequencies never make SHM
  → Adding two SHMs of the same frequency, and telling SHM from periodic motion

Formulas (3)

• Cut springs and springs in series and in parallel · Cut and combined springs
  $$k \propto \frac{1}{l} \qquad \frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2} \qquad k_p = k_1 + k_2$$
• Mass on a spring, and two masses on one spring · Spring–mass period and reduced mass
  $$T = 2\pi\sqrt{\frac{m}{k}} \qquad \omega = \sqrt{\frac{k}{\mu}},\ \ \mu = \frac{m_1m_2}{m_1 + m_2}$$
• SHM from any restoring force or torque · Angular frequency from a restoring force or torque
  $$F = -Cx \Rightarrow \omega = \sqrt{\frac{C}{m}} \qquad \tau = -\kappa\theta \Rightarrow \omega = \sqrt{\frac{\kappa}{I}}$$

Watch out for (9)

• A block between two walls is parallel
  → Cut springs and springs in series and in parallel
• Cutting a spring makes it stiffer
  → Cut springs and springs in series and in parallel
• Gravity along an incline does not change ω
  → Cut springs and springs in series and in parallel
• Two free masses use the reduced mass
  → Mass on a spring, and two masses on one spring
• Gravity does not change a spring's period
  → Mass on a spring, and two masses on one spring
• Equal amplitudes and equal top speeds give opposite ratios
  → Mass on a spring, and two masses on one spring
• A physical pendulum uses I about the pivot
  → SHM from any restoring force or torque
• The tunnel period does not depend on the chord
  → SHM from any restoring force or torque
• A floating body's period depends on its depth under the liquid
  → SHM from any restoring force or torque

Formulas (2)

• Period of a simple pendulum and how it depends on length · Simple pendulum
  $$T = 2\pi\sqrt{\frac{L}{g}} \qquad \frac{\Delta T}{T} = \frac{1}{2}\frac{\Delta L}{L}$$
• Effective g for a pendulum at a height, on a planet, in a lift or in a liquid · Pendulum with an effective g
  $$T = 2\pi\sqrt{\frac{L}{g_{eff}}} \qquad g_h = g\left(\frac{R}{R + h}\right)^{2}$$

Watch out for (7)

• A seconds pendulum has a period of 2 s
  → Period of a simple pendulum and how it depends on length
• The bob's mass does not enter
  → Period of a simple pendulum and how it depends on length
• A longer pendulum makes a clock slow
  → Period of a simple pendulum and how it depends on length
• Large swings: the two accelerations point different ways
  → Period of a simple pendulum and how it depends on length
• A height R puts the pendulum 2R from the centre
  → Effective g for a pendulum at a height, on a planet, in a lift or in a liquid
• The lift's acceleration decides, not its velocity
  → Effective g for a pendulum at a height, on a planet, in a lift or in a liquid
• A clock on a mountain runs slow
  → Effective g for a pendulum at a height, on a planet, in a lift or in a liquid

Formulas (2)

• Kinetic and potential energy at a displacement in SHM · Energy in SHM
  $$E = \frac{1}{2}kA^{2} = \frac{1}{2}m\omega^{2}A^{2} \qquad U = \frac{1}{2}kx^{2} \qquad K = \frac{1}{2}k\left(A^{2} - x^{2}\right)$$
• New amplitude after a push or an added mass, and decay by damping · Amplitude after a sudden change, and damped amplitude
  $$A'^{2} = x^{2} + \frac{v'^{2}}{\omega'^{2}} \qquad A = A_0e^{-bt/2m}$$

Reference tables (1)

Watch out for (8)

• The energies are equal at A/√2, not at A/2
  → Kinetic and potential energy at a displacement in SHM
• A spring's energy does not depend on the mass
  → Kinetic and potential energy at a displacement in SHM
• "Energy of the block" at a point is the total
  → Kinetic and potential energy at a displacement in SHM
• Energy oscillates at twice the frequency
  → Graphs and averages of energy in SHM
• Potential energy is never negative
  → Graphs and averages of energy in SHM
• Where the mass is added decides the new amplitude
  → New amplitude after a push or an added mass, and decay by damping
• Multiply the speed, not the energy
  → New amplitude after a push or an added mass, and decay by damping
• Energy decays twice as fast as amplitude
  → New amplitude after a push or an added mass, and decay by damping