Question

Mass of a particle is 2 kg. Its displacement-time equation in SHM is
$x=2\sin \left(4\pi t\right)$ (SI Units)
Match the following two columns for 1 second time interval :

Answer 1

For the given displacement equation of SHM, we have

$m=2\text{kg}$ and $x=2\sin \left(4\pi t\right)$ and $\omega =4\pi \Rightarrow f=2\text{Hz}$

Thus, in the time period of $1\text{s}$, the system completes $2$ full oscillations.

$\Rightarrow v=\frac{dx}{dt}=8\pi \cos \left(4\pi t\right)$

$\Rightarrow a=\frac{dv}{dt}=-32{\pi }^2\sin \left(4\pi t\right)$

$\text{KE}=\frac{1}{2}m{v}^2=64{\pi }^2{\cos }^2\left(4\pi t\right)$

From the above equations, $v_{\text{max}}=8\pi \approx 25.132$

Thus, $v$ can never be $30\text{m/s}$

In a single oscillation, $v$ takes a value twice. Thus, $v$ becomes $+10\text{m/s}$ for $4$ times in $1\text{s}$

Similarly, ${\text{KE}}_{\text{max}}=64{\pi }^2\approx =631.64$

Thus, the mass attains $\text{KE}$ of $400\text{J}$

In one oscillation, the mass attains this value $4$ times. Hence, in $1\text{s}$, it attains $8$ times.

Similarly, $a_{\text{max}}=32{\pi }^2\approx 315.82$

Thus, similar to velocity, acceleration also attains the value $-100$ for $4$ times in $1\text{s}$