Velocity in SHM

The speed of a particle \(v\) in uniform circular motion is its angular speed \(\omega\) times the radius of the circle \(A\).
\(
v=\omega A \dots(14.8)
\)
The direction of velocity \(\overline{\mathbf{V}}\) at a time \(t\) is along the tangent to the circle at the point where the particle is located at that instant. From the geometry of Fig. 14.11, it is clear that the velocity of the projection particle \(\mathrm{P}^{\prime}\) at time \(t\) is
\(
v(t)=-\omega A \sin (\omega t+\phi) \dots(14.9)
\)

where the negative sign shows that \(v(\mathrm{t})\) has a direction opposite to the positive direction of \(x\)-axis. Eq. (14.9) gives the instantaneous velocity of a particle executing SHM, where displacement is given by Eq. (14.4) which is
\(
x(t)=A \cos (\omega t+\phi)
\)
We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. 14.4) with respect of \(t\).
\(
v(t)=\frac{\mathrm{d}}{\mathrm{d} t} x(t) \dots(14.10)
\)

Acceleration in SHM

The method of reference circle can be similarly used for obtaining the instantaneous acceleration of a particle undergoing SHM. We know that the centripetal acceleration of a particle \(\mathrm{P}\) in uniform circular motion has a magnitude \(v^2 / \mathrm{A}\) or \(\omega^2 \mathrm{~A}\), and it is directed towards the centre i.e., the direction is along PO. The instantaneous acceleration of the projection particle \(P^{\prime}\) is then (See Fig. 14. 12)
\(
\begin{aligned}
a(t) & =-\omega^2 A \cos (\omega t+\phi) \dots(14.11) \\
& =-\omega^2 x(t)
\end{aligned}
\)

Eq. (14.11) gives the acceleration of a particle in SHM. The same equation can again be obtained directly by differentiating velocity \(v(t)\) given by Eq. (14.9) with respect to time:
\(
a(t)=\frac{\mathrm{d}}{\mathrm{d} t} v(t) \dots(14.12)
\)
We note from Eq. (14.11) the important property that the acceleration of a particle in SHM is proportional to displacement. For \(\mathrm{x}(t)>0\), \(a(t)<0\) and for \(x(t)<0, a(t)>0\). Thus, whatever the value of \(x\) between \(-A\) and \(A\), the acceleration \(a(t)\) is always directed towards the centre.

For simplicity, let us put \(\phi=0\) and write the expression for \(x(t), v(t)\) and \(a(t)\)
\(
x(t)=\mathrm{A} \cos \omega t, v(t)=-\omega \mathrm{A} \sin \omega t, a(t)=-\omega^2 \mathrm{~A} \cos \omega t
\)
The corresponding plots are shown in Fig. 14.13. All quantities vary sinusoidally with time; only their maxima differ and the different plots differ in phase. \(x\) varles between \(-A\) to \(A\); \(v(t)\) varles from \(-\omega A\) to \(\omega A\) and \(a(t)\) from \(-\omega^2 A\) to \(\omega^2 A\). With respect to the displacement plot, the velocity plot has a phase difference of \(\pi / 2\) and the acceleration plot has a phase difference of \(\pi\).

Example 1: A body oscillates with SHM according to the equation (in SI units),
\(x=5 \cos [2 \pi t+\pi / 4] \text {. }\)
At \(t=1.5 \mathrm{~s}\), calculate the (a) displacement, (b) speed, and (c) acceleration of the body.

Solution:

The angular frequency \(\omega\) of the body \(=2 \pi \mathrm{s}^{-1}\) and its time period \(T=1 \mathrm{~s}\).
At \(t=1.5 \mathrm{~s}\)
(a) displacement \(=(5.0 \mathrm{~m}) \cos \left[\left(2 \pi \mathrm{s}^{-1}\right) \times\right.\) \(1.5 \mathrm{~s}+\pi / 4]\)
\(=(5.0 \mathrm{~m}) \cos [(3 \pi+\pi / 4)]\)
\(=-5.0 \times 0.707 \mathrm{~m}\)
\(=-3.535 \mathrm{~m}\)
(b) Using Eq. (14.9), the speed of the body
\(
\begin{aligned}
& =-(5.0 \mathrm{~m})\left(2 \pi \mathrm{s}^{-1}\right) \sin \left[\left(2 \pi \mathrm{s}^{-1}\right) \times 1.5 \mathrm{~s}\right. \\
& +\pi / 4] \\
& =-(5.0 \mathrm{~m})\left(2 \pi \mathrm{s}^{-1}\right) \sin [(3 \pi+\pi / 4)] \\
& =10 \pi \times 0.707 \mathrm{~m} \mathrm{~s}^{-1} \\
& =22 \mathrm{~m} \mathrm{~s}^{-1}
\end{aligned}
\)
(c) Using Eq.(14.10), the acceleration of the body
\(
\begin{aligned}
& =-\left(2 \pi \mathrm{s}^{-1}\right)^2 \times \text { displacement } \\
& =-\left(2 \pi \mathrm{s}^{-1}\right)^2 \times(-3.535 \mathrm{~m}) \\
& =140 \mathrm{~m} \mathrm{~s}^{-2}
\end{aligned}
\)