Time period, frequency, and amplitude

Simple Harmonic Motion (SHM) is a type of oscillatory motion where the restoring force is proportional to displacement and acts in the opposite direction.
The time period (T) of SHM is the time taken to complete one full oscillation.
The frequency (f) is the number of oscillations completed in one second.
The relationship between time period and frequency is given by f = 1/T.
The amplitude (A) is the maximum displacement of the oscillating particle from its equilibrium position.
The angular frequency (ω) is related to the frequency by ω = 2πf.
The time period of a mass-spring system is given by T = 2π√(m/k), where m is the mass and k is the spring constant.
The frequency of a mass-spring system is f = (1/2π)√(k/m).
The time period of a simple pendulum is T = 2π√(l/g), where l is the length and g is the acceleration due to gravity.
The frequency of a simple pendulum is f = (1/2π)√(g/l).
The amplitude determines the maximum energy of the system but does not affect the time period or frequency in SHM.
The unit of time period is seconds (s), and the unit of frequency is hertz (Hz).
The unit of amplitude depends on the nature of motion, e.g., meters (m) for linear motion.
The time period of SHM is independent of amplitude for ideal systems.
The velocity of the particle in SHM is maximum at the equilibrium position and is given by v = ωA.
The maximum acceleration in SHM occurs at the extreme positions and is given by a = ω²A.
The displacement in SHM varies sinusoidally with time, following x = A sin(ωt + φ) or x = A cos(ωt + φ).
The amplitude is always a positive quantity and represents the extent of oscillation.
The total energy in SHM is proportional to the square of the amplitude, i.e., E = 1/2 kA².
In SHM, the angular frequency depends on the system's properties, such as mass and stiffness, but not on amplitude.
The frequency of oscillation increases if the system's stiffness (k) increases or the mass (m) decreases.
The amplitude of oscillation can be increased by supplying more energy to the system.
In a mass-spring system, the amplitude is determined by the initial displacement or force applied.
The time period is affected by external factors such as gravity (for pendulums) but remains constant for a given setup.
In the case of a pendulum, the time period increases with an increase in its length.
The time period of a pendulum is inversely proportional to the square root of gravitational acceleration (g).
The frequency of SHM is directly proportional to the square root of stiffness (k) in a spring system.
The angular frequency ω determines the speed of oscillation and is measured in radians per second.
The amplitude determines the maximum potential energy of the system.
The phase constant φ affects the initial position and direction of motion in SHM but does not influence the time period or frequency.
The motion is periodic, and the time period remains constant, even for different amplitudes in ideal SHM.
The frequency of oscillation is an important parameter for analyzing systems undergoing resonance.
The graph of displacement, velocity, or acceleration with time in SHM is sinusoidal, with a peak corresponding to amplitude.
The angular frequency is derived from the system's natural characteristics and defines its oscillatory behavior.
Changes in amplitude do not affect the frequency or time period, making SHM unique among oscillatory motions.
In real systems, damping reduces the amplitude over time but does not immediately affect the frequency.
The energy in SHM alternates between kinetic and potential forms while remaining constant in total.
The velocity and acceleration are zero at the extreme positions, while the amplitude represents maximum displacement.
The amplitude can be experimentally determined from the peak displacement of the oscillating particle.
The time period for rotational SHM depends on the moment of inertia and restoring torque.
The concept of amplitude, frequency, and time period is essential for understanding wave motion and oscillatory systems.
The frequency is inversely proportional to the time period, emphasizing their reciprocal relationship.
In systems with small damping, the time period remains nearly constant over several oscillations.
Understanding the interrelationship between time period, frequency, and amplitude is crucial for designing oscillatory systems.