Question

The amplitude and time period in an SHM is $0.5cm$, $0.4$ respectively. If the initial phase is $\dfrac{r}{2}$ radian, equation of SHM will be
$\begin{align}
  & a)y=0.5\sin (\dfrac{5}{\pi }) \\
 & b)y=0.5\sin (\dfrac{4}{\pi }) \\
 & c)y=0.5\sin (\dfrac{2.5}{\pi }) \\
 & d)y=0.5\cos (5\pi t) \\
\end{align}$

Answer 1

Hint: Write down the equation of shm in terms of amplitude, time period and the phase angle of the SHM. Next, substitute the given values and we can easily get the equation of SHM. As we don’t know the frequency value, convert the frequency in terms of time period of the motion.

Formula used:
$y=A\sin (\omega t+\phi ) $
$ \omega =\dfrac{2\pi }{T} $

Complete step by step answer:
Let’s write down the given terms,
$T=0.4,A=0.5,\phi =\dfrac{\pi }{2}$
Substituting them in the above equation, we get,

$\begin{align}
  & y=A\sin (\omega t+\phi ) \\
 & y=0.5\sin (\dfrac{2\pi }{T}t+\dfrac{\pi }{2}) \\
 & y=0.5\sin (5\pi t+\dfrac{\pi }{2}) \\
 & y=0.5\cos 5\pi t \\
\end{align}$

So, the correct answer is “Option D”.

Additional Information: Simple harmonic motion is a special type of periodic motion where the restoring force on the object moving g is directly proportional to the magnitude of displacement of the object and is always directed towards the equilibrium position of the object. In simple terms, the repetitive movement back and forth through an equilibrium or central position such that the maximum displacement on either side of the position are equal. Also, the time interval for each completion of vibration will be the same in both cases. The net force in simple harmonic motion is directed proportional to the displacement and in the opposite direction of the displacement made by the object. There are many types of simple harmonic motion. Damped harmonic motion is when the motion of an oscillator is reduced due to some other external force. In this case, the energy of the oscillator dissipates continuously. In the case of small damping, the oscillator remains approximately periodic.

Note: The net force in the simple harmonic motion is always negative because the force acts in the opposite direction of the displacement. The period of the motion is directly proportional to the mass of the system. The motion of the body is always periodic in simple harmonic motion.