Question

A particle having mass 10 g oscillates according to the equation x = (2.0 cm) sin [(100 $s^{-}$${}^1$ t + $\pi$/6]. Find.
(a) the amplitude, the time period and the force constant
(b) the position, the velocity and the acceleration at t = 0.

Answer 1

Given, $x=\left(2.0cm\right)sin[100t+\frac{\pi }{6}]$

$\left(a\right)$ $\therefore$ Amplitude$=2.0cm$,

Angular Velocity , $\omega =100$ $rad/sec$, $\Rightarrow \text{Time Period}=\frac{2\pi }{\omega }=0.02\pi$ $s$,

Force constant, $K=m{\omega }^2=10\times {10}^{-3}\times {100}^2=100$.

$\left(b\right)$ At time, $t=0$

Position $x=2sin\left(\frac{\pi }{6}\right)=2\times \frac{1}{2}=1cm$

Velocity $v=2\times 100cos(100t+\frac{\pi }{6})$,

At $t=0$, Velocity ,$v=200cos\left(\frac{\pi }{6}\right)=100\sqrt{3}cm{s}^{-1}$,

Acceleration $a=-2\times {100}^{2}sin(100t+\frac{\pi }{6})$

At $t=0$, Acceleration $a=-2\times {100}^{2}sin\left(\frac{\pi }{6}\right)=-1\times {10}^{4}cm{s}^{-2}$