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Physics with Calculus: Oscillations and Simple Harmonic Motion

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  • What is simple harmonic motion (SHM)?

    SHM is periodic motion where the restoring force is directly proportional to displacement and acts toward equilibrium, producing oscillations like a mass on a spring.

  • What is the formula for the restoring force in a mass-spring system?

    The restoring force is given by Hooke's Law: \(F_s = -kx\), where k is the spring constant and x is displacement.

  • How is angular frequency ω related to mass and spring constant in SHM?

    \(\omega = \sqrt{\frac{k}{m}}\), where k is the spring constant and m is the mass.

  • Define amplitude, period, and frequency in SHM.

    Amplitude (A): max displacement from equilibrium.
    Period (T): time for one complete oscillation.
    Frequency (f): number of oscillations per second, \(f=\frac{1}{T}\).

  • Write the general solution for displacement in SHM.

    \(x(t) = A \cos(\omega t + \phi)\), where A is amplitude, ω angular frequency, and φ phase constant.

  • How do velocity and acceleration vary in SHM?

    Velocity: \(v(t) = -\omega A \sin(\omega t + \phi)\)
    Acceleration: \(a(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t)\)

  • What is the period of a simple pendulum for small angles?

    \(T = 2\pi \sqrt{\frac{L}{g}}\), where L is pendulum length and g is acceleration due to gravity.

  • What approximation is used for small-angle pendulum oscillations?

    For small angles, \(\theta \approx \sin\theta \approx \tan\theta\), simplifying the pendulum's motion to SHM.

  • How is the moment of inertia I defined for a point mass?

    \(I = mr^2\), where m is mass and r is distance from pivot.

  • What is the equation of motion for a torsional pendulum?

    \(I \frac{d^2 \theta}{dt^2} = -\kappa \theta\), where κ is torsional constant and I moment of inertia.

  • How is angular frequency ω related to torsional constant and moment of inertia?

    \(\omega = \sqrt{\frac{\kappa}{I}}\)

  • What is the total mechanical energy in ideal SHM?

    \(E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2\), constant and sum of potential and kinetic energy.

  • What effect does damping have on oscillations?

    Damping causes amplitude to decrease over time, energy is lost to nonconservative forces, and motion is no longer ideal SHM.

  • Define resonance in forced oscillations.

    Resonance occurs when the driving frequency matches the system's natural frequency, causing large amplitude oscillations.

  • Write the amplitude formula for a driven damped oscillator.

    \(A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega_d^2)^2 + (b\omega_d/m)^2}}\), where F₀ is driving force amplitude, ω₀ natural frequency, ω_d driving frequency, and b damping constant.

  • How is the frequency related to period?

    \(f = \frac{1}{T}\), frequency is the reciprocal of the period.

  • What is the physical meaning of phase constant φ in SHM?

    φ determines the initial position and velocity of the oscillator at time zero.

  • How does uniform circular motion relate to SHM?

    The projection of uniform circular motion onto one axis is SHM, with displacement \(x(t) = A \cos(\omega t + \phi)\).

  • What is the restoring torque in a physical pendulum?

    \(\tau = -mgL \sin\theta\), where L is distance from pivot to center of mass.

  • How is the period of a physical pendulum calculated?

    \(T = 2\pi \sqrt{\frac{I}{mgL}}\), where I is moment of inertia about pivot.