Oscillations: Pendulum and Spring

• Oscillations explore the parameters affecting the period of oscillations for a simple pendulum and spring pendulum.
• The aim is to find the free-fall acceleration (g, m/s²) and spring constant (k, N/m).

Basics of Oscillations

• Oscillation in physics refers to the repetitive back-and-forth movement of an object around a central point or between two states.
• Examples include the swinging of a pendulum, vibrations of a guitar string, and a spring-mass system.

Types of Oscillations

• Oscillations can be classified based on the type of energy, type of motion, and external factors involved.

Oscillations by Type of Energy

• Mechanical oscillations involve the conversion between kinetic and potential energy (elastic or gravitational).
• The restoring force is due to physical properties like elasticity or gravity.
• Examples include a mass-spring system (elastic potential energy) or a simple pendulum (gravitational potential energy).
• Electrical oscillations occur in systems where energy alternates between electric and magnetic fields.
• These typically involve inductors (L) and capacitors (C) in circuits, creating oscillating currents and voltages.
• Electromagnetic oscillations involve the oscillation of electromagnetic fields, where energy propagates as electromagnetic waves.
• Energy alternates between electric and magnetic fields as the wave moves through space.

Oscillations by Type of Motion:

• Translation (linear) oscillations occur when the displacement of the oscillating object follows a straight line.
• The motion is typically harmonic, like simple harmonic motion (SHM).
• An example includes a mass-spring system moving back and forth along a straight path, or pendulum with small angular displacements.
• Angular (rotational) oscillations involve the rotation of an object about a fixed axis.
• The restoring torque acts to bring the system back to equilibrium.
• Displacement is measured in terms of angular displacement (θ).
• An example is a torsion pendulum.

Oscillations Based on External Factors:

• Free oscillations: motion is determined solely by the internal restoring force and the system's properties. For example, a pendulum set into motion and then left to swing freely, or a mass-spring system in a vacuum.
• Damped oscillations: resistive forces like friction or air resistance act on the system, leading to a gradual loss of energy over time.
• Forced oscillations: an external periodic force continuously drives the system, with the external force supplying energy to sustain the motion, counteracting any damping present.
• Resonant oscillations: a special type of forced oscillation where the external driving force's frequency matches the system's natural frequency.
• Resonance occurs when an external factor (like a periodic force) synchronizes with the natural frequency.

Heartbeats and Oscillations

• Heartbeats occur in a rhythmic and periodic manner, characteristic of oscillatory motion.
• Heartbeat is not a simple harmonic oscillation because external factors regulate frequency and amplitude.
• Heartbeat oscillation is damped due to energy loss in the system.
• The heartbeat is a damped, nonlinear biological oscillation regulated by physiological control systems to maintain homeostasis.

Simple Pendulum

• Length (L) is the distance from the pivot point to the center of mass of the pendulum's bob.
• Period (T) is the time it takes for the pendulum to complete one full cycle of oscillation.
• The formula for the period of a pendulum is T = 2π√(L/g).
• Angle of displacement (θ) is the angle between the pendulum's string and the vertical at its maximum displacement.
• For small angles (θ ≤ 15°), the motion can be approximated as simple harmonic motion (SHM).
• Angular natural frequency: ω₀ = √(g/L)
• Free fall acceleration g = 9.81 m/s².

Harmonic Motion (SHM)

• SHM is a type of oscillatory motion where a restoring force acts on an object and is directly proportional to the displacement of the object from its equilibrium position, but in the opposite direction.
• Examples of SHM include a mass-spring system, simple pendulum (small angles), and vibrating strings.

Energy and Oscillations

• In undamped oscillations, the internal energy remains constant.
• In damped oscillations, energy dissipates due to frictional force.

Damped Oscillations in Medicine

• Heart Rate Variability: The heart exhibits damped oscillatory behavior after disruptions, with the heart rate gradually returning to its resting state.
• Vibration Response of Bones and Tissues: Oscillatory tests produce damped responses, with vibrations decreasing in amplitude due to the viscoelastic properties of bones and tissues.
• Blood Pressure Regulation: Blood pressure returns to normal through a damped oscillation process, with the cardiovascular system employing feedback mechanisms to stabilize pressure.

Undamped Oscillations in Medicine

• Pacemaker-Induced Heartbeats: An artificial pacemaker induces regular, undamped oscillations in the heart's rhythm.
• Ultrasound Imaging: Oscillations of the piezoelectric crystal inside the ultrasound probe generate continuous, undamped sound waves.
• MRI Radiofrequency Pulses: Undamped oscillations are used to generate radiofrequency pulses in MRI to excite hydrogen nuclei.

Medical Applications of Oscillations

• Heartbeat and Cardiac Rhythms: The heart functions as a biological oscillator, with oscillations used in electrocardiograms (ECG/EKG) and pacemakers.
• Respiratory System: Breathing follows a rhythmic oscillatory pattern, with ventilators using controlled oscillations and high-frequency oscillatory ventilation (HFOV).
• Human Gait: Walking involves the oscillatory motion of the body, with legs swinging forward and back, creating a rhythmic oscillation.