pendulum. L g = ( . ) u Example . What is the length of a simple pendulum, which ticks seconds ?
Answer From Eq. ( . ), the time period of a simple pendulum is given by, L g = From this relation one gets, gT L The time period of a simple pendulum, which ticks seconds, is s. Therefore, for g = .
m s – and T = s, L is – . (m s ) (s ) × = m ⊳ SUMMARY . The motion that repeats itself is called periodlic motion. .
The period T is the time reequired for one complete oscillation, or cycle. It is related to the frequency v by, v The frequency ν of periodic or oscillatory motion is the number of oscillations per unit time. In the SI, it is measured in hertz : hertz = Hz = oscillation per second = 1s – . In simple harmonic motion (SHM), the displacement x ( t ) of a particle from its equilibrium position is given by, x ( t ) = A cos ( ω t + φ ) (displacement), in which A is the amplitude of the displacement, the quantity ( ω t + φ ) is the phase of the motion, and φ is the phase constant .
The angular frequency ω is related to the period and frequency of the motion by, πν (angular frequency). . Simple harmonic motion can also be viewed as the projection of uniform circular motion on the diameter of the circle in which the latter motion occurs. .
The particle velocity and acceleration during SHM as functions of time are given by, v ( t ) = – ω A sin ( ω t + φ ) (velocity), a ( t ) = – ω A cos ( ω t + φ ) = – ω x ( t ) (acceleration), Thus we see that both velocity and acceleration of a body executing simple harmonic motion are periodic functions, having the velocity amplitude v