Important Terms In Simple Harmonic Motion - Practice Questions & MCQ

1. Amplitude:- 

We know that displacement of a particle in SHM is given by:

$x=A \operatorname{Sin}(\omega t+\phi)$

The quantity A is called the amplitude of the motion. It is a positive constant which represents the magnitude of the maximum displacement of the particle from mean position in either direction.

2. Time period:-

• In SHM, a particle repeats its motion after a fixed interval of time. And this time interval after which the particle repeats its motion is called time period. It is denoted by T.
• Time period is also defined as the time taken to complete one oscillation. And after one time period, both displacement and velocity of the particle are repeated.
• We know that:-

$
x=\operatorname{ASin}(\omega t+\phi)
$

If a motion is periodic with a period $T$, then the displacement $x(t)$ must return to its initial value after one period of the motion; that is, $x(t)$ must be equal to $x(t+T)$ for all $t$ and velocity $v(t)$ must also return to its initial value, i.e., $v(t)$ must be equal to $v(t+T)$. So,

$
\begin{aligned}
& x(t)=x(t+T) \\
\Rightarrow & A \operatorname{Sin}(\omega t+\phi)=A \operatorname{Sin}[\omega[t+T]+\phi] \\
\Rightarrow & \operatorname{Sin}(\omega t+\phi)=\operatorname{Sin}[\omega[t+T]+\phi]
\end{aligned}
$

And

$
\begin{aligned}
& v(t)=v(t+T) \\
\Rightarrow & A \omega \operatorname{Cos}(\omega t+\phi)=A \omega \operatorname{Cos}[\omega[t+T]+\phi] \\
\Rightarrow & \operatorname{Cos}(\omega t+\phi)=\operatorname{Cos}[\omega[t+T]+\phi]
\end{aligned}
$

As we know both Sine and Cosine function repeats itself when their argument increases by $2 \pi$,i.e.,

$
\begin{aligned}
& \omega t+\phi+2 \pi=\omega(t+T)+\phi \\
& \Rightarrow 2 \pi=T \omega \\
& \Rightarrow T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{k}}
\end{aligned}
$

$
\text { where } k=\text { force or spring constant and } m=\text { mass }
$

- Time period can also be written as:-

$
T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{m}{\frac{\text { Force }}{\text { displacement }}}}=2 \pi \sqrt{\frac{m \times \text { displacement }}{m \times \text { acceleration }}}
$
 

$
\Rightarrow T=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}
$

3. Frequency:-

The reciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion.
- It is denoted by f .

$
\begin{aligned}
& f=\frac{1}{T}=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}} \\
& \Rightarrow \omega=2 \pi f ; \text { where } \omega \text { is angular frequency }
\end{aligned}
$

- The unit of frequency is $s^{-1}$ or $\operatorname{Hertz}(\mathrm{Hz})$.
4. Phase:-
. The quantity $(\omega t+\Delta \phi)$ is called the phase.
- It determines the status of the particle in simple harmonic motion.
- If the phase is zero at a certain instant, then:

$
x=A \operatorname{Sin}(\omega t+\phi)=0 \text { and } v=A \omega \operatorname{Cos}(\omega t+\phi)=A \omega
$
 

which means that the particle is crossing the mean position and is going towards the positive direction. 

             Fig:- Status of the particle at different phases

5. Phase constant:-
- The constant $\phi$ is called the phase constant (or phase angle).
- The value of $\phi$ depends on the displacement and velocity of the particle at $t=0$ or we can say the phase constant signifies the initial conditions.
- Any instant can be chosen as $\mathrm{t}=0$ and hence the phase constant can be chosen arbitrarily.