Oscillations, pendulums, and springs

Oscillation is the repetitive motion of a system about its equilibrium position.
Simple Harmonic Motion (SHM) is a special type of oscillation where the restoring force is proportional to displacement and acts in the opposite direction.
A system undergoing SHM exhibits a periodic motion with constant frequency and time period.
The motion of a simple pendulum and a mass attached to a spring are classic examples of SHM.
The equation of motion for SHM is F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement.
For a mass-spring system, the angular frequency is given by ω = √(k/m), where m is the mass.
The time period of a mass-spring system is T = 2π√(m/k).
The frequency of oscillation is the reciprocal of the time period, f = 1/T.
The total energy in SHM remains constant, alternating between kinetic energy (KE) and potential energy (PE).
For a simple pendulum, the time period is T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.
A simple pendulum performs SHM for small angular displacements (θ ≤ 15°).
The motion of a pendulum depends on its length and the value of g at its location.
At the extreme positions of SHM, the velocity of the oscillating object is zero, and the acceleration is maximum.
At the equilibrium position, the velocity is maximum, and the acceleration is zero.
In a mass-spring system, the spring's elastic potential energy is given by PE = 1/2 kx².
The kinetic energy in the mass-spring system is given by KE = 1/2 mv².
The total energy of the system is E = 1/2 kA², where A is the amplitude.
The displacement in SHM is described as x = A sin(ωt + φ) or x = A cos(ωt + φ), where φ is the phase constant.
The motion of a pendulum is affected by air resistance, which leads to damping over time.
A system is said to undergo undamped oscillations when there is no energy loss.
The restoring force in SHM ensures that the system returns to its equilibrium position.
The time period of a pendulum increases with altitude due to the decrease in gravitational acceleration.
The frequency of oscillation of a mass-spring system is independent of amplitude.
The phase constant φ determines the initial position and direction of motion in SHM.
The velocity of the oscillating body in SHM is v = ω√(A² - x²).
In a spring-block system, the spring constant k determines the stiffness of the spring.
Oscillations are faster for systems with higher stiffness (greater k) or lower mass (smaller m).
The graph of displacement, velocity, or acceleration with time in SHM is sinusoidal.
The potential energy and kinetic energy vary cyclically, remaining in phase opposition in SHM.
A physical pendulum behaves like a simple pendulum, with its time period given by T = 2π√(I/mgh), where I is the moment of inertia.
Resonance occurs when a system oscillates with maximum amplitude due to an external periodic force matching its natural frequency.
The concept of SHM is vital for understanding mechanical, electrical, and optical oscillatory systems.
For a pendulum, the time period is independent of the mass of the bob.
The amplitude in SHM determines the maximum displacement but does not affect the time period.
In real systems, energy loss due to friction or resistance leads to damped oscillations.
For a spring system, the force constant k is measured in newtons per meter (N/m).
The time period of SHM is constant and depends only on the system's natural properties.
Coupled oscillations occur when two or more oscillatory systems interact.
The maximum acceleration in SHM is given by a = ω²A, occurring at the extreme positions.
The motion of a pendulum and a spring system serves as a model for studying waves and oscillations.
SHM is characterized by constant angular frequency, irrespective of the amplitude.
The energy stored in a spring during oscillation is proportional to the square of its displacement.
The damping of a pendulum can be reduced by operating it in a low-resistance medium.
The natural frequency of a spring system increases with higher stiffness (k) or reduced mass (m).
Understanding SHM is critical for analyzing resonance, vibrations, and sound waves.