SIMPLE HARMONIC MOTION
`color{blue} ✍️` Consider a particle oscillating back and forth about the origin of an x-axis between the limits + A and – A as shown in Fig. 14.3.
`color{blue} ✍️` This oscillatory motion is said to be simple harmonic if the displacement x of the particle from the origin varies with time as
`color{blue} ✍️` where `A, ω` and `φ` are constants.
`color{blue} ✍️` Thus, simple harmonic motion (SHM) is not any periodic motion but one in which displacement is a sinusoidal function of time. Fig. 14.4 shows what the positions of a particle executing SHM are at discrete value of time, each interval of time being T/4 where T is the period of motion.
`color{blue} ✍️`Fig. 14.5 plots the graph of `x` versus `t,` which gives the values of displacement as a continuous function of time. The quantities A, ω and φ which characterize a given SHM have standard names, as summarised in Fig. 14.6. Let us understand these quantities.
`color{blue} ✍️`The amplitutde A of SHM is the magnitude of maximum displacement of the particle. [Note, A can be taken to be positive without any loss of generality].
`color{blue} ✍️`As the cosine function of time varies from `+1` to `–1`, the displacement varies between the extremes `A` and `– A`. Two simple harmonic motions may have same `ω` and `φ` but different amplitudes A and B, as shown in Fig. 14.7 (a).
`color{blue} ✍️`While the amplitude A is fixed for a given SHM, the state of motion (position and velocity) of the particle at any time t is determined by the argument `(ωt + φ)` in the cosine function. This time-dependent quantity, `(ωt + φ)` is called the phase of the motion.
`color{blue} ✍️`The value of plase at `t = 0` is `φ` and is called the phase constant (or phase angle). If the amplitude is known, φ can be determined from the displacement at `t = 0`. Two simple harmonic motions may have the same `A` and `ω` but different phase angle `φ,` as shown in Fig. 14.7 (b).
`color{blue} ✍️`Finally, the quantity `ω` can be seen to be related to the period of motion `T`. Taking, for simplicity, `φ = 0` in Eq. (14.4), we have
`color{blue} ✍️`Since the motion has a period `T, x(t)` is equal to `x (t + T)`. That is,
`color{blue} ✍️`Now the cosine function is periodic with period `2π`, i.e., it first repeats itself when the argument changes by `2π`. Therefore,
`color{purple} {ω(t + T ) = ωt + 2π}`
that is
`color{blue} ✍️` `ω` is called the angular frequency of SHM. Its S.I. unit is radians per second. Since the frequency of oscillations is simply `1//T, ω` is 2π times the frequency of oscillation.
`color{blue} ✍️`Two simple harmonic motions may have the same A and `φ`, but different `ω`, as seen in Fig. 14.8. In this plot the curve (b) has half the period and twice the frequency of the curve (a).
`color{blue} ✍️` This oscillatory motion is said to be simple harmonic if the displacement x of the particle from the origin varies with time as
`color{blue} {x (t) = A cos (ω t + φ)}`
........................ (14.4)`color{blue} ✍️` where `A, ω` and `φ` are constants.
`color{blue} ✍️` Thus, simple harmonic motion (SHM) is not any periodic motion but one in which displacement is a sinusoidal function of time. Fig. 14.4 shows what the positions of a particle executing SHM are at discrete value of time, each interval of time being T/4 where T is the period of motion.
`color{blue} ✍️`Fig. 14.5 plots the graph of `x` versus `t,` which gives the values of displacement as a continuous function of time. The quantities A, ω and φ which characterize a given SHM have standard names, as summarised in Fig. 14.6. Let us understand these quantities.
`color{blue} ✍️`The amplitutde A of SHM is the magnitude of maximum displacement of the particle. [Note, A can be taken to be positive without any loss of generality].
`color{blue} ✍️`As the cosine function of time varies from `+1` to `–1`, the displacement varies between the extremes `A` and `– A`. Two simple harmonic motions may have same `ω` and `φ` but different amplitudes A and B, as shown in Fig. 14.7 (a).
`color{blue} ✍️`While the amplitude A is fixed for a given SHM, the state of motion (position and velocity) of the particle at any time t is determined by the argument `(ωt + φ)` in the cosine function. This time-dependent quantity, `(ωt + φ)` is called the phase of the motion.
`color{blue} ✍️`The value of plase at `t = 0` is `φ` and is called the phase constant (or phase angle). If the amplitude is known, φ can be determined from the displacement at `t = 0`. Two simple harmonic motions may have the same `A` and `ω` but different phase angle `φ,` as shown in Fig. 14.7 (b).
`color{blue} ✍️`Finally, the quantity `ω` can be seen to be related to the period of motion `T`. Taking, for simplicity, `φ = 0` in Eq. (14.4), we have
`color{blue} {x(t ) = A cos ωt}`
...................... (14.5)`color{blue} ✍️`Since the motion has a period `T, x(t)` is equal to `x (t + T)`. That is,
`color{blue} {A cos ωt = A cos ω (t + T )}`
.................... (14.6)`color{blue} ✍️`Now the cosine function is periodic with period `2π`, i.e., it first repeats itself when the argument changes by `2π`. Therefore,
`color{purple} {ω(t + T ) = ωt + 2π}`
that is
`color{blue} {ω = 2π//T}`
....................(14.7)`color{blue} ✍️` `ω` is called the angular frequency of SHM. Its S.I. unit is radians per second. Since the frequency of oscillations is simply `1//T, ω` is 2π times the frequency of oscillation.
`color{blue} ✍️`Two simple harmonic motions may have the same A and `φ`, but different `ω`, as seen in Fig. 14.8. In this plot the curve (b) has half the period and twice the frequency of the curve (a).
