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:

$x=A\mathrm{Sin}(\omega t+\varphi )$

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:-

$$x=A\mathrm{Sin}(\omega t+\varphi )$$

$$\begin{array}{rl}& x(t)=x(t+T)\\ & \Rightarrow A\mathrm{Sin}(\omega t+\varphi )=A\mathrm{Sin}[\omega [t+T]+\varphi ]\\ & \Rightarrow \mathrm{Sin}(\omega t+\varphi )=\mathrm{Sin}[\omega [t+T]+\varphi ]\end{array}$$

And

$$\begin{array}{rl}& v(t)=v(t+T)\\ \Rightarrow & A\omega \mathrm{Cos}(\omega t+\varphi )=A\omega \mathrm{Cos}[\omega [t+T]+\varphi ]\\ \Rightarrow & \mathrm{Cos}(\omega t+\varphi )=\mathrm{Cos}[\omega [t+T]+\varphi ]\end{array}$$

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

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

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

• - Time period can also be written as:-

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

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 $(\omega t+\mathrm{\Delta }\varphi )$ 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:

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

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 $\varphi$ is called the phase constant (or phase angle).
- The value of $\varphi$ 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.