Important Terms In Simple Harmonic Motion MCQ - Practice Questions with Answers

1. Amplitude:- 

We know that the 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 that represents the magnitude of the maximum displacement of the particle from the mean position in either direction.

2. Time period:-

• In SHM, a particle repeats its motion after a fixed interval of time. 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. After one time period, both displacement and velocity of the particle are repeated.
• We know that:-

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

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

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:-

  $$
  \begin{aligned}
  & T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{m}{\frac{\text { Force }}{\text { displacement }}}}=2 \pi \sqrt{\frac{m \times \text { displacement }}{m \times \text { acceleration }}} \\
  & \Rightarrow T=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}
  \end{aligned}
  $$
   

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.

$$
f=\frac{1}{T}=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}
$$

$\Rightarrow \omega=2 \pi f$; where $\omega$ is angular frequency
- The unit of frequency is $s^{-1}$ or $\mathrm{Hertz}(\mathrm{Hz})$.

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 $=0$ 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.